Clifford Group Equivariant Neural Networks
Authors: David Ruhe
Johannes Brandstetter
Patrick Forré
Venue: NeurIPS 2023 oral (2023)
Keywords: Clifford algebras
geometric deep dearning
Clifford group equivariance
E(n)-equivariant neural networks
O(n)-equivariant neural networks
TL;DR: A method to construct E(n)- and O(n)-equivariant neural networks using Clifford algebras.

Abstract: We introduce Clifford Group Equivariant Neural Networks: a novel approach for constructing $\mathrm{O}(n)$- and $\mathrm{E}(n)$-equivariant models. We identify and study the *Clifford group*: a subgroup inside the Clifford algebra tailored to achieve several favorable properties. Primarily, the group's action forms an orthogonal automorphism that extends beyond the typical vector space to the entire Clifford algebra while respecting the multivector grading. This leads to several non-equivalent subrepresentations corresponding to the multivector decomposition. Furthermore, we prove that the action respects not just the vector space structure of the Clifford algebra but also its multiplicative structure, i.e., the geometric product. These findings imply that every polynomial in multivectors, including their grade projections, constitutes an equivariant map with respect to the Clifford group, allowing us to parameterize equivariant neural network layers. An advantage worth mentioning is that we obtain expressive layers that can elegantly generalize to inner-product spaces of any dimension. We demonstrate, notably from a single core implementation, state-of-the-art performance on several distinct tasks, including a three-dimensional $n$-body experiment, a four-dimensional Lorentz-equivariant high-energy physics experiment, and a five-dimensional convex hull experiment.

Clifford Group Equivariant Neural Networks
David Ruhe A14Science Lab, AMLab, API University of Amsterdam david. roweidiaah@gmai.com com
Johannes Brandstetter
Microsoft Research A14Science
brandstetterGml. jku. .k
Patrick Forré Al4Science Lab, AMLab University of Amsterdam
p.d.forre@uva.r nl
Abstract
We introduce Clifford Group Equivariant Neural Networks: novel approach for constructing O(n)- and E(n)- -equivaraa models. We identify and study the Clifford group: subgroup inside the Clifford algebra tailored to achieve several fa- vorable properties. Primarily, the group's action forms an orthogonal automorphism that extends beyond the typical vector space to the entire Clifford algebra while respecting the multivector grading. This leads to several non- eouiiiuiall subrepre- sentations corresponding to the multivector decomposition. Furthermore, we prove that action respects not just the vector space structure the Clifford algebra but also its multiplicative structure, i.e., the geometric product. These findings imply that every polynomial multivectors, including their grade projections, constitutes equivariant map with respect to the Clifford group, allowing to parameter- ize equivariant neural network layers. An advantage worth mentioning is that we obtain expressive layers that can elegantly generalize to inner- menrrprruuctt spaces any dimension. We demonstrate, notably from a single core implementation, state-of-the of-the-art performance several distinct tasks, including three-dimensional n- n-body experiment, a cour-dimensional Lorentz-equivariant high-energy physics experiment, and a five-dimensional convex hull experiment.
Introduction
Incorporating group equivariance ensure symmetry constraints in neural networks has been a highly
fruitful line of research [CWI6 WGTB17 CGKWI8 KT18 WC19 WGW + 18 Est20 BBCV21
WFVW2I CLW22] Besides translation and permutation equivariance ZKR 17 SGT 08 rotation equivariance proved to be vitally important for many graph- -trph-structured problems encountered in, e.g., the natural sciences. Applications of such methods include modeling the dynamics of complex physical systems or motion trajectories KFW- 18, BHP+ 22]; studying generating molecules, SSK 18, ZCD+20 TVS+21 21 AGB22] ; and point cloud analysis [WSK+1s UPH+19 19). Note that many of these focus on three-dimensional
proteins, and crystals IDRL14 GLB+16 16 CTS
problems involving rotation, reflection, or translation equivariance by considering the groups O(3),
SO(3), E(3), or SE(3).
Such equivariant neural networks can be broadly divided into three categories: approaches that scalarize geometric quantities, methods employing regular group representations, and those utilizing irreducible representations, often of ((3) [RXH222. Scalarization methods operate exclusively on
is available http:///wwwggg.ccc......coaaatte
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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+2
e13 e20
9123
p(w)
Scalars
Vectors
Bivectors
Trivectors
((w)
Figure I: CGENNs (represented with Φ) are able to operate on multivectors (elements the Clifford algebra) in an Onn)- or E(n) eqquivaraaa way. Specifically, when an action p(w) of the Clifford group, representing an orthogonal transformation such as a rotation, is applied to the data, the model's representations corotate. Multivectors can be decomposed into scalar, vector, bivector, trivector, and even higher-order components. These elements can represent geometric quantities such as (oriented) areas volumes. The action p(w) is designed to respect these structures when acting on them.
scalar features or manipulate higher-order geometric quantities such vectors via scalar multiplica- tion ISSK CCGIS, KGG20 KKN20 SHW21 JES+21 GBG21 SUG21 DLD+21 HHR +22
TF22. They can be limited by the fact that they do not extract all directional information. Regular representation methods construct equivariant maps through an integral over the respective group (CW16]KTI) For continuous Lie groups, however, this integral is intractable and requires coarse approximation FSIW20 Bek19]. Methods of the third category employ the irreducible repre- sentations of O(3) (the Wigner-D matrices) and operate in steerable spherical harmonics basis AHKI9 FWFW20 BHP 22 BMS 22]. This basis allows a decomposition into type- vector subspaces that transform under D': the type-/ matrix representation of 0(3) [en90 FA 91]. Through tensor products decomposed using Clebsch-Gordan coefficients (Clebsch- Gordan tensor product), vectors (of different types) interact equivariantly. These tensor products can be parameter- ized using learnable weights. Key limitations of such methods include the necessity for an alternative basis, along with acquiring the Clebsch-Gordan coefficients, which, although they are known for unitary groups of any dimension [KR67], are not trivial to obtain [AKHvD11].
[TSK 18
We propose Clifford Group Equivariant Neural Networks (CGENNs): an equivariant parameterization of neural networks based on Clifford algebras. Inside the algebra, we identify the Clifford group and its action, termed the (adjusted) twisted conjugation, which has several advantageous properties. Unlike classical approaches that represent these groups on their corresponding vector spaces, we carefully extend the action to the entire Clifford algebra. There, it automatically acts an orthogonal automorphism that respects the multivector grading, enabling nontrivial subrepresentations that operate on the algebra subspaces. Furthermore, the twisted conjugation respects the Clifford algebra's multiplicative structure, i.e. the geometric product, allowing us to bypass the need for explicit tensor product representations.
a result, we obtain two remarkable properties. First, all polynomials in multivectors generate Clifford group equivariant maps from the Clifford algebra to itself. Additionally, grade projections are equivariant, allowing for denser parameterization of such polynomials. We then demonstrate how to construct parameterizable neural network layers using these properties.
Our method comes with several advantages. First, instead of operating on alternative basis repre- sentations such as the spherical harmonics, CGENNs (similarly to scalarization methods) directly transform data vector basis. Second, multivector representations allow (geometrically mean- ingful) product structure while maintaining a finite dimensionality as opposed to tensor product representations. Through geometric products, we can transform vector-valued information, resulting in more accurate and nuanced interpretation of the underlying structures compared to scalarization methods. Further, we can represent exotic geometric objects such as pseudovectors, encountered in certain physics problems, which transform in nonstandard manner. Third, our method readily generalizes to orthogonal groups regardless of the dimension or metric signature of the spuce, thereby attaining ((n)- or E(n)-co -uuivaquiva These advantages are demonstrated on equivariance bench-

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marks of different dimensionality. Note that specialized tools were developed for several of these tasks, while CGENNs can be applied more generally.
2 The Clifford Algebra
Clifford algebras (also known as geometric algebras) are powerful mathematical objects with ap- plications in various areas of science and engineering. For a complete formal construction, we refer the reader Appendix D Let V be finite-dimensional vector space over field F equipped with a quadratic form q V= F. The Clifford algebra CI(V, q) is the unitary, associative, non- commutative algebra generated by V such that for every V the relation v² q(v) holds, i.e., vectors square to scalars. This simple property solely generates a unique mathematical theory that underpins many applications. Note that every element æ of the Clifford algebra CI(V, q) is linear combination of (formal, non-commutative) products of vectors modulo the condition that every appearing square v² gets identified with the scalar square q(v): x Eier C: V4,1 ks Here, the index set I is fimite, C E F, k E No, Va.j E V. The Clifford algebra's associated bi- linear form b(U1, U2) ;= 1 (q(v) + v2) ((v1)=
( 1) yields the fundamental Clifford identity: 2122 + 12 2b( 22) for U1, U2 V (Lemma D.3 In this context, the quantity U112 repre- sents the geometric product, which is aptly named for its ability to compute geometric properties and facilitate various transformations. Note that when 11, 1/2 are orthogonal (e.g., for orthogonal basis vectors), b(21.22 222) = 0, in which case U1 = 11 The dimensionality of the algebra is 2n, where n == dim V (Theorem D.15). Let 61, ......, C, be an orthogonal basis of V. The tu- ple [m)) [m] = {1...., n). is an orthogonal basis for CI(V.q). where for all such A the product A I 6; is taken in increasing order (Theorem D.26 We will see below that we can decompose the algebra into vector subspaces (T) VV.4).m m = 0, ....., n, called grades, where dim (1(m) (V.q)= = (*). Elements of grade m = 0 and m = 1 are scalars (CI
F) and vectors (V q) = V), respectively, while elements of grade m = 2 and m = 3 are referred to as bivectors and trivectors. Similar terms are used for elements of even higher grade. These higher-order
grades can represent (oriented) points, areas, and volumes, as depicted in Figure] |
Clifford algebras provide tools that allow for meaningful algebraic representation and manipulation of geometric quantities, including areas and volumes [RDK21] DM02 DFM09). In addition, they offer generalizations such as extensions of the exterior and Grassmannian algebras, along with the natural inclusion of complex numbers and Hamilton's 's quaternions (Gra62 [am66). Applications of Clifford algebras can be found in robotics [BCRI ZE06 HZZC08)) computer graphies [WCL05)
BTH22), signal processing 12it BMQQS+21
and animation [HPI0, [C12). In the context of machine learning, Clifford algebras and hypercomplex numbers have been employed to improve the performance of algorithms in various tasks. For example, [MFW21] learn an equivariant embedding using geometric neurons used for classification tasks. Further, [Spe2T] introduce geometric algebra attention networks for point cloud problems in physics, chemistry. and biology. More recently, [BBWG22] introduce Clifford neural layers and Clifford Fourier transforms for accelerating solving partial differential equations. [RGdK 23] continue this direction. strengthening the geometric inductive bias by the introduction of geometric templates. Concurrently with this work, [BDHBC23] develop the geometric algebra transformer. Further, [LK14, PRM 18, ZXXCI8] introduce complex- valued and quaternion-value networks. Finally, [KIdHN23] define normalizing flows on the group of unit quaternions for sampling molecular crystal structures.
(HP2)
3 Theoretical Results
In order to construct equivariant multivector-vall neural networks, we outline our theoretical results. We first introduce the following theorem regarding the multivector grading of the Clifford algebra, which is well-known in case the algebra's 's metric is non-degenerate. Although the general case, including a potentially degenerate metric, appears to be accepted, we were unable to find self-contained proof during our studies. Hence, we include it here for completeness.
Theorem 3.1 (The multivector grading of the Clifford algebra). Let Ei...., en and bi...., bn be two orthogonal bases of (V, q). Then the following sub-vector spaces (1(m) (V, q) of CI(V. q).
If k; 0 the product of vectors Vi.1 0i,k, is empty, and, in this case, we mean the unit in CI(V. q).
3

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m O,..... are independent of the choice of the orthogonal basis, i.e.,
11 (V, q) == span {ea | C [n], |A| = m) - span {bA A C |n], =m..
(I)
The proof can be found in Theorem D.27 Intuitively, this means that the claims made in the following (and the supplementary material) are not dependent on the chosen frame of reference, even in the degenerate setting. We now declare that the Clifford algebra CI(V. q) decomposes into an orthogonal
direct sum of the vector subspaces CI (m)) (V, q). To this end, we need to extend the bilinear form b from V to CI(V, q). For elements 3 22, E CI(V, q), the extended bilinear form b and the extended quadratic form q are given via the projection onto the pro-component (explained below), where 3: CI(V, q) + CI(V, q) denotes the main anti-involution of CI(V, 9
(2) Note that by construction, both 6 and q reduce to their original versions when restricted to V. Using the extended quadratie form, the tuple (CI q). q) turns into quadratic vector space itself. As a corollary (see Cor3llaryD.30)). the Clifford algebra has an orthogonal-sun decomposition w.r.t. the extended bilinear form b:
6 CI(V.q) q) x q)(V.q) FF
((2) = )x,)))²)))
q(x) := 6(2 x).
CI(V, q) = C1(m) (V,9).
dim C1 (m) (V.q) = 12
(3)
m==
This result implies that can always write z E CI(V, q) as = z0)) + x (2) + x x(1,, where (m(m) E (1(m) (V.q) denotes the grade-m part of æ. Selecting grade defines an orthogonal projection:
( (mm) CI(V, q) + (1 (V, g),
x + (m))
m = 0, 0,...,?? n.
(4)
Let us further introduce the notation C1 (V.q) == θ mh even parity, and (V, q) == t add (l (V, q) for those of odd parity. We use = [0 + x! to denote the parity decomposition of a multivector.
(l(m) (V. whose elements are of even
3.1 The Clifford Group and its Clifford Algebra Representations
Let CI* (V, q) denote the group of invertible elements of the Clifford algebra, i.e., the set of those ele-
ments w E CI(V, q) that have an inverse w-1 CI(V, q): w--w ww 1. For w E C1* (V, we then define the (adjusted) twisted conjugation as follows:
(5) where is the main involution of CI(V, g). which is given by a(w) == wl - w This map p(w)= : CI(V. q) + CI(V. g). notably not just from V V, will be essential for constructing equivariant neural networks operating on the Clifford algebra. In general, p and similar maps defined in the literature do not always posses the required properties (see Motivation E.T.. However, when our p is restricted a carefully chosen subgroup of CI* (V,q), many desirable characteristics emerge. This subgroup will be called the Clifford grou of CI(V, q) and we define it as:
p(w)= CI(V, 4) CI(V,)) 9),
((w)(x) == wo f 1 + a(w) w
I(V,a) == {w C1 (V, n (c q) U (V.a) Ve E V, ((w)(o) E
(6) In words, [(V, q) contains all invertible (parity) homogeneous elements that preserve vectors (m = 1 elements) via p. The special Clifford group is defined r (V.q) I(V.q) n (V.
Regarding the twisted conjugation, p(w) was ensured to reduce to reflection when restricted to V a property that we will conveniently use in the upcoming section. Specifically, when CI*(V, V, w E CI* (V, 4). then p(w) reflects in the hyperplane normal w:
w, x: E
p(w)(x) www _
b(w, x))
b(w, w)
Next, we collect several advantageous identities of p in the following theorem, which we elaborate on afterwards. For proofs, consider Lemma E.8 Theorem E.10 and Theorem E.29
Recall that any J CI(V, q) can be written linear combination of vector products. is the map that reverses the order: 8 (Eiez G5 88,1 ... v.,... := E.er & Us,b, ... U111. For details, see Definition D.18 We elaborate shortly the term Clifford group in contrast similar definitions in Remark E.14

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é times
(
CI(V,q) x .... CI(V.q)
F
CI(V,q)
CI(V,9)
Cl(m)
(V,9)
p(w) (o(w)
p(w)
plw)
((w)
p(w)
(m)
CI(V,9) q) x.... x CI(V.q)
CI(V,q)
CI(V,9)
(7(m) (V,9)
Figure 2: Commutative diagrams expressing Clifford group equivariance with respect to the main operations: polynomials F (left) and grade projections (m)) (right).
Theorem 3.2. Let wi, W2, w P(V,)). 22, x CI(V,q), c C E, m E no. p then satisfies:
1. Additivity: p( 12) = ((w) (22)
2. Multiplicativity; ((w)(( p(w)(=))((\(()))
and: p(w)(c) c,
3. Invertibility: ))-))({) (r))
4. Composition: p(w2) (((w)((x)) p (z) and: p(c)(z) for + 0.
5. Orthogonality: b (p(w)(mn), (x2)) 321,
The first two properties state that p(w) not only additive, but even multiplicative regarding the geometric product. The third states that p(w) is invertible, making it an algebra automorphism of CI(V,q), q). The fourth property then states that p I(V.q) + Aut A (CI(V. (VV. q)) is also a group homomorphism to the group of all algebra automorphisms. In other words, CI(V.q 1) is a linear representation of (VV 9) and, moreover, last point shows that each p(w) generates an orthogonal map with respect to the extended bilinear form These properties yield the following results (see also Theorem E.16 Corollary E.18 Figure 2).
also an algebra representation of [(V, q). Finally, the
Corollary 3.3 (All grade projections are Clifford group equivariant). For w E [(V, q). x E CI(V,
and O, ..... have the following equivariance property:
(lw)(x((m)) = (x)
(8)
In particular, for (1(m) (V,9) also have ((w) E (1) (1(m)(((1))) q).
This implies that the grade projections: CI(V.q) + (1 (V, q) are [(V. -)-eqvivaria maps, and, that each (m) (V, q) constitutes orthogonal representation of T(V, q). The latter means that p induces a group homomorphisms from the Clifford group to the group of all orthogonal invertible linear transformations of (m) (V, (V,q):
(m) I(V.q) + (CI (V.4).a),
(w) = p(
Corollary 3.4 (All polynomials are Clifford group equivariant). Let E F[T...., Te] be a polynomial in E variables with coefficients in E, I(V.q). Further, consider i elements F1, II,..., Dt E CVVV, q). Then we have the following equivariance property:
p(w) (F (F(Z
P(olw))sss.....
(10)
To prove that p(w) distributes over any polynomial, we use both its additivity and multiplicativity regarding the geometric product.
By noting that one can learn the coefficients such polynomials, we can build flexible parameteriza- tions of Clifford group equivariant layers for quadratic vector spaces of any dimension or metric. We involve grade projections to achieve denser parameterizations. More details regarding neural network constructions follow in Section
5

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3.2 Orthogonal Representations
To relate to the orthogonal group O(V, q), the set of invertible linear maps : V - V preserving q. first note that Equation (5) shows that for every w E IF we always have: p(w) = idci(V.a)- So the action p on CI(V, q) can already be defined on the quotient group T(V. 4)/Rx. Moreover, we have:
Theorem 3.5 (See also Remark F.31 and Corollary E.32 ). (V, q) is non-degenerate then induces
well-defined group isomorphism:
p T(V,4))P² - O(V.)),
[w] + p(w)lv-
(11)
The above implies that O(V, q) acts whole CI(V, q) in well-defined way. Concretely, if a E CI(V,q) q) is of the form æ = Eier G Vi,1 Mg with wiis E V. G E and & E O(V,q) is given by ([w]) = & with w E q), then we have:
p(w)(z)= = Σ a . p(w)(()))) ((w)( E c ((1,1) ET
2 (v,x))
(12)
iET
This means that & acts on a multivector x by applying an orthogonal transformation to all its vector components vi,j. as illustrated in Figure []] Theorem 3.5 implies that that map f: CI(V, - CI(V, is equivariant to the Clifford group T(V, q) if and only if it is equivariant to the orthogonal group O(V, q) when both use the actions on CI(V, q) described above (for each component). To leverage these results for constructing (V, -)qqqiaari neural networks acting vector features, i.e., V'n, we refer to Section
To prove Theorem 3.5 we invoke Cartan Dirta--Dieudonéé (Theorem C.13). stating that every orthogonal transformation decomposes into compositions of reflections, and recall that reduces to a reflection when restricted to V, see Theorem E.25 Theorem 3.5 also allows one to provide explicit descriptions
of the element of the Clifford group, see Corollary F.27
Finally, it is worth noting that our method is also Pin and Spin group-equivarian These groups, sharing properties with the Clifford group, are also often studied in relation to the orthogonal group.
4 Methodology
We restrict our layers to use F = TR. Our method is most similar to steerable methods such as [BHP=22] 22]] However, unlike these works, we do not require alternative basis representation based on spherical harmonics, nor do we need to worry about Clebsch-Gordan coefficients. Instead, we consider simply a steerable vector basis for V, which then automatically induces steerable multivector basis for CI(V, q) and its transformation kernels. By steerability, we mean that this basis can be transformed in a predictable way under action from the Clifford group, which acts orthogonally on both V and CI(V, q) (see Figure D].
We present layers yielding Clifford group-equivariant optimizable transformations. All the main ideas are based on Corollary 3.3 and Corollary (3.4 It is worth mentioning that the methods presented here form a first exploration of applying our theoretical results, making future optimizations rather likely.
Linear Layers ZI. Bg E CI(V, q) be a tuple of multivectors expressed in a steerable basis, where l represents the number of input channels. Using the fact that polynomial restricted to the first order constitutes a linear map, we can construct a linear layer by setting
y Coee
Tlin (x1,
DE) (8)
(13)
where denacak E IR are optimizable coefficients and Cin- Cour denote the input and output channel, respectively. As such, T : CI(V.q)* - CI(V, q) is a linear transformation in each algebra subspace
'Here restrict (V, q) be non-degenerate as we never consider degenerate quadratic forms in experiments. However, the supplementary material we generalize this also to the degenerate case by carefully taking the radical subspace R of (V. q) into account on both sides of the isomorphism.
We discuss in Appendix E.6 general definition of these groups

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k. Recall that this is possible due to the result that p(w) respects the multivector subspaces. This computes a transformation for the output channel Cours the map can repeated (using different sets of parameters) for other output channels, similar to classical neural network linear layers. For yeur E IR (the scalar part of the Clifford algebra), we can additionally learn invariant bias parameter.
Geometric Product Layers A core strength of our method comes from the fact that we can also parameterize interaction terms, In this work, we only consider layers up second order. Higher-order interactions are indirectly modeled via multiple successive layers. As an example, we take the pair (k)
21, T2. Their interaction terms take the form use of the fact that p(w) respects grade projections. As such, all the grade-k terms resulting from the interaction of y and 22 are parameterized with
,i,j O..... n; where we again make
Pa(21, (k)
(6) (1) Digk 11 22
)
(14)
where Pa: CI(V,)) q) x CI(V,q) - CI(V, q). This means that we get (n + 1)a parameters for every geometric product between a pair of multivectors] Parameterizing and computing all second-order terms amounts to ( such operations, which can be computationally expensive given reasonable number of channels f. Instead, we first apply a linear map to obtain yı....,, yr E CI(V, 9). Through this map, the mixing (i.e.. the terms that will get multiplied) gets learned. That is, we only get & pairs (I1, 3/), ..., (Ie, yc) from which we then compute zédu == Porn (Ze,, Ve..) (80) Note that here have Gn Cout. ie., the number of channels does not change. Hence, we refer to this layer as the element-wise geometric product layer. We can obtain a more expressive (yet more expensive) parameterization by linearly combining such products by computing
(1))
7 prod
Te, 11 ................................ Me (k))
P Poenay (Ieur Mea. (00)
(15)
Pesat
which call the fully-connected geometric product layer. Computational feasibility and experimental verification should determine which parameterization is preferred.
Normalization and Nonlinearities Since our layers involve quadratic and potentially higher-order interaction terms, we need to ensure numerical stability. In order to do so, we use normalization operating on each multivector subspace before computing geometric products by putting
((m)
(1))
(16)
(am)
(q(x(())) - 1) +1
where (m)) (V Here, a denotes the logistic sigmoid function, and a. a learned scalar. The denominator interpolates between 1 and the quadratic form q (m)( normalizing the magnitude of ((!)) This ensures that the geometric products do not cause numerical instabilities without losing information about the magnitude of z where learned scalar interpolates between both regimes. Note that by Theorem 3.2 (()(()) is invariant under the action of the Clifford group, rendering Equation (6) equivariant map.
Next, we use the lader-wide normalization scheme proposed by [RGdK +3), which, since is also based on the extended quadratic form, is also equivariant with respect to the twisted conjugation.
Regarding nonlinearities,
slightly adjusted version of the units proposed by IRGdK+23
Since the scalar subspace (1 (V.q) always invariant with respect to the twisted conjugation, we can apply mm(m) + ReLU (m)) when m 0 and (m)) + 0. (q (z<()) (((m) otherwise. We can replace ReLU with any common scalar activation function. Here, a represents a potentially parameterized nonlinear function. Usually, however, we restrict it to be the sigmoid function. Since we modify (m)) with an invariant scalar quantity, we retain equivariance. Such gating activations are
commonly used in the equivariance literature [WGW+ 18 GS22
II practice, we use fewer parameters due to the sparsity of the geometric product, implying that many interactions will invariably be zero, thereby making their parameterization redundant.
7

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E(n)-MLP
GVP
MLP+Aug EMLP SO(5)
EMLP O(5)
VN
MLP
CGENN (Ours)
10-1
30
10a
10
20
109
10-a
10
10-2
10-7
10-4
10 0(3) Signed Volumes
10
10a 10*
10a
10
10*
((5) Convex Hulls
0(5) Regression
Figure 3: Left: Test mean-squared errors on the ((3) signed volume task as functions of the number of training data. Note that due to identical performance, some baselines are not clearly visible. Right: same, but for the (55) convex hull task.
Figure
Test mean-
squared-errors on the ((5) regression task.
Embedding Data in the Clifford Algebra In this work, consider data only from the vector space V or the scalars IR, although generally one might also encounter, e.g., bivector data. That is, have some scalar features ....... hu E R and some vector features hg+1,---. he V. Typical examples of scalar features include properties like mass, charge, temperature, and so on, Additionally, one-hot encoded categorical features are also included because {0, 1} I and they also transform trivially. Vector features include positions, velocities, and the like. Then, using the identifications (1 (V. q) z IR and (1 (V, q) @ V, we can embed the data into the scalar and vector subspaces of the Clifford algebra to obtain Clifford features .1,...,. It CI(V,9) 9).
Similarly, we can predict scalar- or vector-valued data as output of our model by the scalar or vector parts, respectively. We can then directly compare these quantities with ground- truth vector- or scalar-valued data through a loss function and use standard automatic differentiation techniques optimize the model. Note that invariant predictions are obtained by predicting scalar quantities.
onto
5 Experiments
Here, we show that CGENNs excel across tasks, attaining top performance in several unique contexts. Parameter budgets as well as training setups are kept as similar as possible to the baseline references. All further experimental details can be found in the public code release.
5.1 Estimating Volumetric Quantities
((3) Experiment: Signed Volumes This task highlights the fact that equivariant architectures based on scalarization are not able to extract some essential geometric properties from input data. In synthetic setting, simulate dataset consisting of random dimenssiia tetrahedra. A main advantage of our method is that can extract covariant quantities including (among others) signed volumes, which we demonstrate in this task. Signed volumes are geometrically significant because they capture the orientation of geometric objects in multidimensional spaces. For instance, in computer graphics, they determine whether 3D object is facing towards away from the camera, enabling proper rendering. The input to the network is the point cloud and the loss function the mean-squared error between the signed volume and its true value. Note that we are predicting a covariant (as opposed to invariant) scalar quantity (also known as pseudoscalar) under O(3) transformations using a positive-defintt (Euclidean) metric. The results are displayed in the left part of Figure 3 We compare against standard multilayer perceptron (MLP), an MLP version of the E(n)-GNN SHW21 which uses neural networks update positions with scalar multiplication, Vector Neurons (VN) [DD* 21]. and Geometric Vector Perceptrons (GVP) (JES 21). We see that scalarization methods fail to access the features necessary for this task, as evidenced by their test loss not improving even with more available data. The multilayer perceptron, although a universal approximator, lacks the correct inductive biases. Our model, however, has the correct inductive biases (c.g., the equivariance property) and can also access the signed volume. Note that we do not take

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the permutation invariance of this task into account, as we are interested in comparing our standard architectures against similar baselines.
(55) Experiment: Convex Hulls We go step further and consider a five-dimennsioal Euclidean
space, showcasing our model's ability to generalize to high dimensions. We also make the experiment more challenging by including more points and estimating the volume of the convex hull generated by these points a task that requires sophisticated algorithms in the classical case. Note that some points may live inside the hull and do not contribute the volume. We use the same network architectures as before (but now embedded in a five-dimensional space) and present the results in Figure 3 We report the error bars for CGENNs, representing three times the standard deviation of the results of eight runs with varying seeds. Volume (unsigned) is an invariant quantity, enabling the baseline methods to approximate its value. However, we still see that CGENNs outperform the other methods, the only exception being the low-data regime of only 256 available data points. We attribute this to our method being slightly more flexible, making slightly more prone to overfitting. To mitigate this issue, future work could explore regularization techniques or other methods to reduce overfitting in low-data scenarios.
5.2 ((5) Experiment: Regression
We compare against the methods presented by [FWW21] who propose an ((5)-invarian) regression problem. The task to estimate the function [z1, x2) == sin([|=;,I)) ||> 3 + TT Illlll P2
x,_ 22
where
the five-dimensional vectors I1, I2 are sampled from a standard Gaussian distribution in order to simulate train, test, and validation datasets. The results are shown in Figure We used baselines from [FW21 including an MIP,, MLP with augmentation (MI.P+Au)), and the ((5)- & (OL5)-MLP architectures. We maintain the same number of parameters for all data regimes. For extremely small datasets (30 and 100 samples), we observe some overfitting tendencies that can be countered with regularization (e.g.. weight decay) or using a smaller model. For higher data regimes (300 samples and onward), CGENNs start to significantly outperform the baselines.
5.3
E(3) Experiment: n-Body System
The n-body experiment KFWW 18] serves as a benchmark for assessing the performance of equivariant (graph) neural networks in simulating physical systems [HRXH22). In this experiment, the dynamics of n three-dimensional space are simulated. Given the initial posi- tions and velocities of these particles, the task is to accurately estimate their positions after 1 0000 timesteps. To address this challenge, we construct graph neural network (GNN) using the Clifford equivariant layers introduced in the previous sec- tion. We use a standard message-passing algorithm |GSR* + 17 where the message and update networks are CGENNs. So long as the message aggregator is equivariant, the end-to-end model also maintains equivariance. The input to the network consists of the suear-subbe positions of the particles (to achieve translation invariance) and their velocities. The model's 's output is the estimated displacement, which is to the input to achieve translation -qquivaaann estimated target positions. We include the invariant charges as part of the input and their products as edge attributes. We compare against the steerable Transforrssssssmrs [FWFW20], Tensor Field Networks [TSK+ + 18), and SEGNN [BHP+22] Scalarization basclines include Radial Field [KKN20] and EGNN [SHW21]. Finally, NMP [GSR+ is not an E(3)- cquivariant method. The number of parameters in our model is maintained similar to the EGNN and SEGNN baselines to ensure a fair comparison.
Method
MSE (1)
SE(3)-Tr.
0.0244
TFN
0.0155
charged particles in
NMP
0.0107
Radial Field
0.0104
EGNN
0.0070
SEGNN
0.0043
CGENN
0.0039 + 0.0001
Table 1: Mcan-squared error (MSE) on the n-body system experiment.
Results of our experiment are presented in Table where we also present for CGENN three times the standard deviation of three identical runs with different seeds. Our approach clearly outperforms earlier methods and significantly better than [BHP+22), thereby surpassing the baselines. This experiment again demonstrates the advantage of leveraging covariant information in addition to scalar quantities, as it allows for a more accurate representation of the underlying physics and leads to better predictions.

---

1/EB (T) (Es = 0.5)
1/EB (1)
Model
Accuracy (1)
AUC (f)
(Eg 0.3)
ResNeXt IXGD 17
0.936
0.9837
302
1147
P-CNN CMS17
0.930
0.9803
201
759
PFN [MTT99
0.932
0.9819
247
888
ParticleNet [QG20)
0.940
0.9858
397
1615
EGNN [SHW21]
0.922
0.9760
148
540
LGN [BAO 20
0.929
0.9640
124
435
NorentzNet [GMZ*
0.942
0.9868
2195
CGENN
0.942
0.9869
500
2172
Table 2:
Performance comparison between proposed method and alternative algorithms the top tagging experiment. We present the accuracy, Area Under the Receiver Operating Characteristic Curve (AUC), and background rejection 1/cB and at signal efficiencies of es 0.3 and cs 0.5.
5.4 (11,3) Experiment: Top Tagging
Jet tagging in collider physics a technique used to identify and categorize high-energy jets produced in particle collisions, as measured by, e.g., CERN's ATLAS detector [BTB 08). By combining information from various parts of the detector, it is possible to trace back these jets origins [KNS ATL17]. The current experiment seeks tag jets arising from the heaviest particles of the standard model: the "top quarks' [[QWW09] A jet tag should be invariant with respect to the global reference frame, which can transform under Lorentz boosts due the relativistic nature of the particles. A Lorentz boost is a transformation that relates the space and time coordinates of an event as seen from two inertial reference frames. The defining characteristic of these transformations is that they preserve the Minkowski metric, which is given by (ct,, 2, z) == (ct)² - x² y²² - z² Note the difference with the standard positive definite Euclidean metric, used in the previous experiments. The set of all such transformations is captured by the orthogonal group ( 3); therefore, our method fully compatible with modeling this problem.
We evaluate our model on a top tagging benchmark published by [KPTRI9] It contains 1 1.2M training entries, 400k validation entries, and 400k testing entries. For each jet, the energy momentum 4-vectors are available for up to 200 constituent particles, making this a much larger-scale experiment than the ones presented earlier. Again, employ standard message passing graph neural network [GSR+ + using CGENNs as message and update networks. The baselines include ResNeXt [XGD+17) P- CNN (CMSI7), PFN [KMT19] ParticleNet (QG20). LGN [BAO+20), EGNN (SHW21), and the more recent LorentzNet G2]++22) Among these, LGN is steerable method, whereas EGNN and LorentzNet scalarization methods. The other methods not Lorentz-equivvaiaatt Among the performance metrics, there are classification accuracy, Area Under the Receiver Operating Characteristic Curve (AUC), and the background rejection rate 1/CB signal efficiencies of es 0.3 and cs = 0.5, where Eg and are the false positive and true positive rates, respectively. We observe that LorentzNet, method that uses invariant quantities, an extremely competitive baseline that was optimized for this task. Despite this, CGENNs are able to match its performance while maintaining the same core implementation.
Conclusion
We presented novel approach for constructing ((n)- and Clifford algebras. After establishing the required theoretical results, we proposed parameterizations of nonlinear multivector- valued maps that exhibit versatility and applicability across scenarios varying in dimension. This was achieved by the core insight that polynomials in multivectors are O(n)- equivariant functions. Theoretical results were empirically substantiated in three distinct experiments, outperforming or matching baselines that were sometimes specifically designed for these tasks.
neural networks based on
CGENNs induce (non- prooibrohibitive) degree of computational overhead similar to other steerable methods. On the plus side, we believe that improved code implementations such as custom GPU kernels alternative parameterizations to the current ones can significantly alleviate this issue, potentially also resulting in improved performances on benchmark datasets. This work provides solid theoretical and experimental foundations for such developments.
10

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Acknowledgments
We thank Leo Dorst and Steven de Keninck for insightful discussions about Clifford (geometric) algebras and their applications as well as pointing us to the relevant literature. Further, we thank Robert-Jan Schlimbach and Jayesh Gupta for their help with computational infrastructure and scaling up the experiments. This work used the Dutch national e-infrastructure with the support of the SURF Cooperative using grant no. EINF-5757 as well as the DAS-5 and DAS-6 clusters [BEdl 16].
Broader Impact Statement
The advantages of more effective equivariant graph neural networks could be vast, particularly in the physical sciences. Improved equivariant neural networks could revolutionize fields like molecular biology, astrophysics, and materials science by accurately predicting system behaviors under various transformations. They could further enable more precise simulations physical systems. Such new knowledge could drive forward scientific discovery, enabling innovations in healthcare, materials engineering, and our understanding of the universe. However, a significant drawback of relying heavily advanced equivariant neural networks in physical sciences the potential for unchecked errors or biases to propagate when these systems are not thoroughly cross-validated using various methods. This could potentially lead flawed scientific theories dangerous real-world applications.
Reproducibility Statement
We have released the code to reproduce the experimental results, including hyperparameters, network architectures, and so on, at http:://ggthbbbcccm///adddddeiid heapioooddroouuuuuuuuuueeee neural-networks
Computational Resources
The volumetric quantities and regression experiments were carried 1 x 11 GB NVIDIA GeForce GTX 1080 Ti and x GB NVIDIA GeForce GTX 2080 Ti instances. The n-body experiment ran on 1 x 24 GB NVIDIA GeForce RTX 3090 and 1 24 GB NVIDIA RTX A5000 nodes. Finally, the top tagging experiment was conducted on 4 40 GB NVIDIA Tesla A100 Ampere instances.
Licenses
We would like thank the scientific software development community, without whom this work would not be possible.
This work made use of Python (PSF License Agreement), NumPy [VDWCVII] (BSD-3 3 Clause), PyTorch 19 (BSD-3 Clause), CUDA (proprietary license), Weights and Biases (Bie20) (MIT),
Scikit-Learn PPVG 11) (BSD-3 Clause), Seaborn [Was21] (BSD-3 Clause), Matplotlib Huno7
(PSF), [VGO+ 20 (BSD-3).
The top tagging dataset is licensed under CC-By Attribution 4.0 International.
17

---

Contents
Introduction
2
The Clifford Algebra
Theoretical Results
The Clifford Group and its Cliftord Algebra Representations
Orthogonal Representations
Methodology
5
Experiments
5.1
Estimating Volumetric Quantities
8
5.2
O(5) Experiment: Regression
9
5.3
E(3) Experiment: n-Body System
5.4
0(1,3) Experiment: Top Tagging
10
6
Conclusion
10
References
11
Acknowledgements
17
Broader Impact Statement
17
Reproducibility Statement
17
Computational Resources
17
Licences
17
Glossary
20
Supplementary Material: Introduction
22
Quadratic Vector Spaces and Orthogonal Group
23
The Clifford Algebra and Typical Constructions
26
D.1
The Clifford Algebra and its Universal Property
26
D.2 The Multivector Filtration and the Grade
27
D33 The Parity Grading
28
D.4
The Dimension of the Clifford Algebra
29
D.5
Extending the Quadratic Form to the Clifford Algebra
32
D.6
The Multivector Grading
36
D.7
The Radical Subalgehra of the Clifford Algebra
44
D.8
The Center of the Clifford Algebra
45
D.9
The Twisted Center of the Clifford Algebra
48
18

---

E
The Clifford Group and its Clifford Algebra Representations
49
Adjusting the Twisted Conjugation
50
The Clifford Group
54
E.3
The Structure of the Clifford Group
57
Orthogonal Representations of the Clifford Group
60
E.S
The Spinor Norm and the Clifford Norm
63
E.6
The Pin Group and the Spin Group
66
19

---

A
Glossary
Notation
Meaning
(n)
The orthogonal group acting on an n-dimensional vector space.
SO(n)
The special orthogonal group acting on an n-dimensional vector space.
F((n)
The Euclidean group acting n--dimensional vector space.
SE(n)
The special Euclidean group acting
n-dimensional vector space.
GL(n)
The general linear group acting on an n-dimennional vector space.
9(V.g)
The orthogonal group of vector space V with respect to a quadratic form q.
(R(V.q)
The orthogonal group of vector space V with respect quadratic form q that acts as the identity on R.
A vector space.
R
The radical vector subspace of V such that for any E b(f, v) for all o E V.
A field.
The real numbers.
N
The natural numbers.
Z
The integers.
[n]
The set integers {1,...,n1 n3. A quadratic form, q: V - [f. A bilinear form, b: V V + F.
q
6
CI(V,))
The Clifford algebra over a vector space V with quadratic form
CI (V.9)
The group invertible Clifford algebra elements.
(V,
The group invertible harty-hhomggeneuus Clifford algebra elements.
CiLxI (V, == {w CI (V, q) n(w) E {=1)).
(V,
The even subalgebra of the Clifford algebra. C1 (V.q) == D even
CI (V.g).
(V,
The odd part of the Clifford algebra. C1 q) == O add (m (V,q).
(V
The grade-m subspace of the Clifford algebra.
(m))
Grade projection, (L)(m) CI(V, > (1(m)) (V,q).
<
Projection onto the zero grade, Ç CI(V,q) + CI (V.q). A basis vector, E; V.
fo
A basis vector of R. A Clifford basis element (product of basis vectors) with еλ E CI(V,q). g
ea
{l,...,,7) n).
The extended bilinear form on the Clifford algebra, 6 CC(VV q) x CI(V. +
q
The extended quadratic form on the Clifford algebra, q CI(VV.) + F.
Clifford main involution a CI(V, q) - CI(V.q).
Main Clifford anti- involution, also known reversion, 3 CI(V. q) - CI(V.g).
Clifford conjugation 7 CI(V, - CI(V.q).
Coboundary of a. For w E C1 (V, 4), (ww) E (=1} if and only if w is parity- homogeneous.
p(w)
The (adjusted) twisted conjugation, used as the action of the Clifford group. p(w): CI(V, q) + CI(V, q), w e CI (V,q).
I(V,q)
The Clifford group CI(V,q).
20

---

Notation
Meaning
(V.
The special Clifford group. It excludes orientation-reversing (odd) elements. (V,4) = r(V.)) q) n CI (V.q).
A(R)
The radical subalgebra of CI(V, q)). Le., those elements that have zero q. It is equal to the exterior algebra of R.
A 19 (R)
The even or odd (i E {0, 1)) subalgebra of A(R). ( = A(R) n
Cl (V.q).
A (R)
The subalgebra of A(R) with grade greater than or equal to one. A (R)
span ... f& f, R}.
(R)
The group of invertible elements of A(R). (R =F A (R).
A (x] (R)
The group of invertible elements of A(R) with even grades. A [x] (R) = FP + span (ʃi ... fk
even, Si E R).
N (R)
Same < (R), but with Fx set to 1.
(R)
Same as A [x] (R), but with Fx set to 1.
SN
Spinor norm, SN CI(V, q) +
CN
Clifford norm, CN CI(V.q) q) + F.
21

---

B Supplementary Material:
Introduction
Existing literature on Clifford algebra uses varying notations. conventions, and focuses on several different applications. We gathered previous definitions and included independently derived results to achieve the desired outcomes for this work and to provide proofs for the most general cases, including. e.g., potential degeneracy of the metric. As such, this appendix acts as a comprehensive resource on quadratic spaces, orthogonal groups, Clifford algebras, their constructions, and specific groups represented within Clifford algebras.
We start in Appendix C with primer quadratic spaces and the orthogonal group. In particular, we specify how the orthogonal group is defined on quadratic space, and investigate its action, We pay special attention to the case of spaces with degenerate quadratic forms, and what we call "radical-preserving orthogonal automorphisms. The section concludes with the presentation of the Cartan-Dieudonné theorem.
In Appendix D we introduce the definition the Clifford algebra as quotient the tensor algebra. We investigate several key properties of the Clifford algebra. Then, we introduce its parity grading, which in turn is used to prove that the dimension of the algebra is 2", where v is the dimension of the vector space which it is defined. This allows to construct algebra basis. Subsequently, extend the quadratic and bilinear forms of the vector space to their Clifford algebra counterparts. This then leads the construction of an orthogonal basis for the algebra. Furthermore, the multivector grading of the algebra is shown to be basis independent, leading to the orthogonal sum decomposition into the usual subspaces commonly referred as scalars, vectors, bivectors, and so on. Finally, we investigate some additional properties of the algebra, such as its center, its radical subalgebra, and its twisted center.
In Appendix we introduce, motivate, and adjust the twisted conjugation map. We show that comprises an algebra automorphism, thereby respecting the algebra's is vector space and also its multiplicative properties. Moreover, it can serve as a representation of the Clifford algebra's 's group of invertible elements. We then identify what call the Clifford group, whose action under the twisted conjugation respects also the multivector decomposition. We investigate properties of the Clifford group and its action. Specifically, we show that its action yields radical- preservvig orthogonal automorphism. Moreover, it acts orthogonally on each individual subspace.
Finally, after introducing the Spinor and Clifford norm, the section concludes with the pursuit of general definition for the Pin and Spin group, which are used in practice more often than the Clifford group. This, however, turns out to be somewhat problematic when generalizing fields beyond RR We motivate several choices, and outline their (dvs)advansages. For R we finally settle on definitions that are compatible with existing literature.
22

---

C Quadratic Vector Spaces and the Orthogonal Group
We provide a short introduction to quadratic spaces and define the orthogonal group of (non-definite) quadratic space. We use these definitions in our analysis of how the Clifford group relates to the orthogonal group.
We will always denote with a field of a characteristic different from 2, char(IF) of 2. Sometimes we will specialize to the real numbers IF R. Let V be vector space over of finite dimension dimp V = n. We will follow [Ser12].
Definition C.1 (Quadratic forms and quadratic vector spaces). A map q V - will be called quadratic form V for all E F and v V:
q(c v) = q()),
(17)
and if:
b(U1, 02) = (9(0) + v2) - q(1)) - -V2))),
(18)
is a bilinear form over IF in 0i, E V, i.e., it is sepurately FF-lneer in each of the arguments in and 02 when the other one is fixed.
The tuple (V, q) will then be called a quadratic (vector) space.
Remark C.2.
Note that we explicitely do not make assumptions about the non-degeneracy
of b. Even the extreme case with constant q = 0 is allowed and of interest.
2. Further note, that will automatically be symmetric bilinear form.
Definition C.3 (The radical subspace). Now consider the quadratic space (V, q). We then call the subspace:
R: (VV
(19)
the radical subspace of (V.q).
Remark C.4.
1. The radical subspace of (V,q) is the biggest subspace of V where q is degenerate. Note that this space is orthogonal all other subspaces of V.
2. W is any complementary subspace R in V, so that V W is non-degenerate.
R + W, then restricted to
Cn of V is called orthogonal basis of V for all Definition C.5 (Orthogonal basis). A basis 61, ei....;Cr i + we have:
b(e,, ez) 0.
(20)
It is called orthonormal basis if, in addition, q(ei) { (-1,0,+11 for all == 1,...,n
Remark C.6. Note that every quadratic space (V,q) has an orthogonal basis by [Ser121 p. 30 Thm. 1, but not necessarily a orthonormal basis. However, if = IR then (V,q) V) has an orthonormal basis by Sylvester's 's law of inertia, see [Sy152].
Definition C.7 (The orthogonal group). For quadratic space (V, we define the orthogonal group of (V. q) as follows:
O(g) = OVV.g) = { V V + F-linear automorphisn s.t. Ve E V.q(((e)) q(v))
(21)
=n. If (V.g) = ((aar))) L.e. ifF = R and V = R and has the signature (p, r) with + + r= then we define the group of orthogonal matrices of signature (p, q, r) as follows:
O r.) = {0 e GL(n) OT A (pgr.) O = A (p.r))
(22)
where used the (n x n) diagonal signature matrix:
A (p.g.)) := diag +1,
(23)
getims
23

---

Theorem C.8 (See ) Let (V.q) be a finite dimensional quadratic space over a field FF with (F + 2. Let R S V be the radical subspace, r dim R, and, W V complementary subspace, m = dim W. Then we gel an isomorphism:
NOW.alw) M(r,m) m)
Omxr GL(r)
={
{ 0mxr) Ge) o alw), G
9) =
(24)
where M(r, m.) = Frxm is the additive group of all (r mi--aatrices with coefficients in FF and where GL(r) is the multiplicatiive group of all invertible (r x r-)matrices with coefficients in F.
Proof. Let C1,..., be an orthogonal basis for W and fi....; fo be a basis for R, then the associated bilinear form b of q has the following matrix representation:
( 9)
(25)
with an invertible diagonal (m x m))mmmmme matrix Q. For the matrix of any orthogonal automorphism & of V we get the necessary and sufficient condition:
AT
C
Q
0
A
B)
(26)
BT
D
0
C
ATQA ATQB)
(27)
BTQA
B QBQ)
This is equivalent to the conditions:
Q = AAA,
0= A QBQ
0 = QB.
(28)
This shows that is the matrix of an orthogonal automorphism W (w.r.t. b/w), which, since Q is invertible, is also invertible. The second equation then shows that necessarily B = 0. Since the whole matrix needs to be invertible also must be invertible. Furthermore, there are no constraints on
If all those conditions are satisfied the whole matrix satisfies the orthogonality constraints from above.
Corollary C.9, For (P:a.)) we get:
O(p.g,r)=
O(p. ④) M +
((p+g) GL(r)
(29)
where O(p, q) = O(p, 0).
Remark C.10. Note that the composition 4 o of orthogonal automorphisms (V, q) corresponds the matrix multiplication as follows:
000₂ (0)) 2.) (Oz 2.) - -M100MM0 G, G.G.)- )
(30)
Definition C.11 (Radical preserving orthogonal automorphisms). For a quadratic space (V, q) with radical subspace R C V we define the group of radical preserving orthogonal automorphisms as follows:
OR q) = { E O(V.)) 인 = idR..
(31)
If (V,4) = R((aa))) i.e. ifF R and V = nhah and has the signature (p, g,,r,) with + q 4+++ then we define the group of radical preserving orthogonal matrices of signature (p. r) as follows:
OR(p,ar) = {0 O(p. bottom right corner of O =1,} = (O0,))
0(p,g)
0 ( xr
(M(r.++9)
I
(32)
where L, is the (r x rididentity matrix.
24

---

Remark C.12. Note that the matrix representation of t E (R( W.E.T. an orthogonal basis like in the proof of theorem C.8 is of the form:
(Nr
0mxr
(33)
Ir
with O ((W.lw)) M E M(r, m) and where I, is the (r riiidntity matrix.
The composition D1 22 of di and D2 E Ove(g) is then given by the corresponding matrix multiplication:
(oi 2.)
1.)(2O
O2 M2 2.)
(MO
O002 M2
1)
(34)
By observing the left column (the only part that does not transform trivially), that O(qlw) acts on M(r, m. just by matrix multiplication from the right (in the corresponding basis):
(1), (02, M2) = (O,02, Mi0 + M2)-
(35)
This immediately shows that write (g(q) as the semi-direct product:
Oz = ((llw) x M(r,m).
(36)
In the special case of R R(p.g.r) we get:
Or = Opp.)) M(r, 9)-
(37)
We conclude this chapter by citing the Theorem of Cartan and Dieudonné about the structure of orthogonal groups in the non-degnerate case, but still for arbitrary fields IF with char(F) # 2.
Theorem C.13 (Theorem of Cartan-Dieudonaa
[Ar557] Thm. 3.20). Let (V.q) be
degenerate quadratic space of finite dimension dim V = n < 0o over a field F of char (F) * 2. Then every element g O(V, q) can be wrillen
g Tk-
(38)
with S k S n, where T; are reflections w.r.l. non-singular hyperplanes.
25

---

D The Clifford Algebra and Typical Constructions
In this section, we provide the required definitions, constructions, and derivations leading to the results stated in the main paper. We start with general introduction to the Clifford algebra.
D.1 The Clifford Algebra and its Universal Property
We follow LS009 Cru90).
Let (V, be finite dimensional quadratic space field F (also denoted an F-vector space) with
(F * 2. We abbreviate the corresponding bilinear form on vectors 81, U2 V as follows:
b(v1, 02) (q(v) + v2) q(vi) (vv2)).
(39)
Definition D.1. To define the Clifford algebra CI(V, q) we first consider the tensor algebra of
V:
20
T(V)
θ V span
(U1 @ Um m 2 0, 0, VV
(40)
V = F, (41)
and the following two-sided idea
[()) = (u 2 -uu)) (T()) a E v).
(42)
Then we define the Clifford algebra CI(V, 9) as the following quotient:
CI(V, q) := T(V)/((q)
(43)
In words, we identify the square of a vector with its quadratic form. We also denote the canonical algebra quotient map as:
T(V) - CI(V, q).
(44)
Remark D.2.
It is not easy to see, but always true, that dim CI(V, q) 2", where n dim V, see Theorem D.15
2. e1,..... Cn is any basis of (V.g) q) then (eA)ag(a) is a basis for CI(V, q). where we put for
a subset S [n] == (l....,.................. n):
CA =
II Cs1
eg = ICI(V.))
(45)
where the product is taken in increasing order of the indices E A, see Corollary D616
3. If ei, en is any orthogonal basis of (V, 9). then one orthogonal basis for CI(V, q) w.r.t. an extension of the bilinear form b from V to CI(V. q), see Theorem D.26
show that (e AçIni is an
Lemma D.3 (The fundamental identity). Note that for U1, 02 E V, we always have the fundamental identity in CI(V,q):
U1 22 + 02 = 2b(11, v2).
(46)
Proof. By definition of the Clifford algebra we have the identities:
(w1) + 01 + 02 + q(r2) = u1 + 01 + 02 + U212
(47)
(wi v2) + 02)
(48)
=q(1 + 02)
(49)
b(0i U2, 01 + +22
(50)
b(0). vn) + 6(01, v2) + b(02. 01) + 6(02, 02)
(51)
=q(u;) 2b(21, 22) ((2)).
(52)
Substracting ((1)) + ((2)) on both sides gives the claim.
"The ideal ensures that for all elements containing v @ v (i.c., linear combinations, multiplications on the left,
and multiplications on the right), 3 v is identified with q(v): e.g. x v ~ x (q(e) IT(Y)) @ for every 2, y E T(V).
26

---

Further, the Clifford algebra CI(V, q) is fully characterized by the following property.
Theorem D.4 (The universal property of the Clifford algebra). For every E F-algebra aa algebra over fvel F) A and every F-linear map f V + A such that for all v E V we have:
f(u)² ( 1A.
(53)
there exists unique F-algebra homomorphisn v E V.
CI(V,q) A such that f(v) f(v) for all
More explicitely, if f satisfies equation 53 and x E CI(V. q). Then we can take any representation of x of the followingform form:
2 = co +
Σ EET
11 U,,k,
(54)
with fiite index sets I and kis E N and coefficients co. Ct E F and vectors Vi.j E V. = 1, 1,..., ki
i E I, and, then we can compute f(x) by the following formula (without ambiguity):
=( 1A
Σa Ca.((v)))) (.Vd))))))) GET
(55)
the following, we will often denote J again with f without further indication.
D.2 The Multivector Filtration and the Grade
Note that the Clifford algebra is a filtered algebra. Definition D.5. We define the multivector filtration of CI(V, q) for grade No as follows:
(m (V.q) = (T(SM)(V))
T(Sm) = 0ot
(56)
I=
Remark D.6. Note that we really get a filtration on the space CI(V. q):
F (V,4) S ca(su)(V S c)($2)(V, E... (N)) (V.q) = C)(V (57)
Furthermore, note that this is compatible with the algebra structure of CI(V, 4). i.e. for i, 2 0 get:
x E CECI(S) A y (
Ty E CI(Sit) (V,
(58)
Together with the equality: कt (T((S) c1lsm) (V. (V. 9) for all m, we see that the natural map:
k: T(V) CI(V.
(59)
a surjective homomorphism of filtered algebras. Indeed, since CI(V, q) is a well-defined algebra, since we modded out two-sided ideal, π is clearly homomorphism of filtered algebras. As quotient map π is automatically surjective.
Definition D.7 (The grade of an element). For E CI(V,q) \ {0) we define its grade through the following condition:
grd =
such that
(i(s (V, Cl(Sk-I((V.)).
(60)
grd 0 = 00.
(61)
For the purpose of this text an algebra is always considered to be associative and unital (containing an identity element), but not necessarily commutative. An algebra homomorphism is both linear and multiplicative.
A filtration F is an indexed family (Ac)ee (1 is an ordered index set) of subsets of an algebraic structure A such that for i j A: C Aj.
27

---

D.3 The Parity Grading
In this section, we introduce the parity grading of the Clifford algebra. We will use the parity grading to later construct the (adjusted) twisted conjugation map, which will be used as group action on the Clifford algebra.
Definition D.8 (The main involution). The linear map:
V - CI(V, q),
a(v) == =
(62)
satisfies (-o)²= v² qv). The universal property of CI(V, q) thus extends unique algebra homomorphism:
a: CI(V,q) - CI(V,g),
co
Σ
cj
22,1
02,82
(63)
iE!
00
Σ
Ci a(vi,)) ... ..(.E))))))
(64)
iEl
co
+
Σ
1) 1ilkiieee .4 e.....0.. U1,1 U..1 Ui
(65)
iEI
for any finite sum representation with U.j E V and a E This extension a will be called the main
involution T3 of q).
Definition D.9 (Parity grading). The main involution a of CI(V,q) q) now defines the parity grading of CI(V, 9) via the following homogeneous parts:
CI (V.4) = fx CI(V. a(x) (a(z) x},
(66)
aI (V.4) = {- CI(V, --x-.
(67)
With this we get the direct sum decomposition:
CI(V, q) (V, 4) @ (V,4)
(68)
10] + !I
00] (z + a(x)), I!ll 2( a(x)),
(69)
with the homogeneous parts 20 E (V q) and 1 E (
We define the parity of an (homogeneous) element x E CII(V, q) as followws
prt(#) So ifx C1 (V,q): if C1 (V,q).
(70)
Definition D.10 (2/22-graded algebras). An F-algebra (A, +.-) together with a direct sum decom- position of sub-vector spaces:
A A A
(71)
called a 2//2Z-graded algebra for every i.j E 2/22 we always have:
I. E At A
y y AbI
x y Al++l
(72)
Note that [i+ is meant here to be computed modulo 2. The 2/22 -radde of an (homogeneous) element E A will also be called the parity of
iʃ E Alo {
prt(z)=
(73)
if Alll.
The above requirement then implies for homogeneous elements x, y E A the relation:
prt(x y) prt(z) prt(y) mod 2.
(74)
We summarize the results of this section as follows: Theorem D.11. The Clifford algebra CI(V, q) is a graaded algebra in its parity grading.
An involution is map that is its inverse.
28

---

D.4 The Dimension of the Clifford Algebra
In this subsection we determine the dimension of the Clifford algebra, which allows us construct bases for the Clifford algebra.
In the following, again let IF be any field of char(F) * 2.
Definition/Lemma D.12 (The twisted tensor product of 2/22-graded F F-algebras). Let A and B be two -22d-gd algebras over R. Then their twisted tensor product A6sB is defined via the usual tensor product A 69 B of F-vector spaces, but where the product defined on homogeneous elements a1 E.A. b,, b2 E via:
(a1 (ay@b2) = (-1) (-1)ou(b))petla) ((1b2)
(75)
This turns A also into a 22/22-graded algebra over F with the Z/ -/ZZdngrddnn:
(A&B) = (A10) _.o)
(A B!)).
(76)
(A&B) = (Al") 2 B + AA 2 BTn)
(77)
In particular, if A, b E B are homogeneous elements then have:
prt Aa = (r) + prbz(b) mod 2.
(78)
Proof. By definition we already know that A&B is an F-vector space. So we only need to investigate the multiplication and g//2Zggr
For homogenous elements d1, 02; 03 A, b,, b2 E B we have: &b)) (az(b22) (ag(bbs)
(79)
(- -1))n((u))pr(ea))
(- 1)pr ((bbb)) (ag&by)
(80)
(81)
(- 1)ptlbb
(82)
= (-1) p (- pm((s) (0 (010203) (bbbb2b))
(83)
(-1) prt(as) (a: )b)) -rrttts))
((a20)) (b2bs))
(84)
(ajābi) ((a2&b2) (azgbs)).
(85)
This shows associativity of multiplication on homogeneous elements, which extends by linearity to general elements. The distributive law is clear.
To check that we have we have:
note that for homogeneous elements 01, ag E A. b1, b2 E B
A6B (a1 (a2&bz)) = A&t 11
(86)
prt ( + (b
(87)
(a1) + prt (a2) + (b1) + prtg(b2)
(88)
prt A&& (ar + prs (12@b2) mod 2.
(89)
The general case follows by linear combinations.
Remark D.13 (The universal property of the twisted tensor product of Z 22/2Z-graded F Flalebbras). Let A, A2, B be 2/2Z- 22added F-algebras. Consider Z/2Z- graded F- Flleebaa homomorphisms:
v1 : A +B B.
2/2
A + B.
(90)
such that for all homogeneous elements E Ai and a2 E A2 we have:
(1(11) - (11)))/2(000) (02) = (-1) (11))rr((1)-ppt(en) (a2(22) thi (a1).
(91)
Then there exists unique 7./27--rrdde F-algebra homomorphism:
w/: A A2 - B3
(92)
29

---

such that:
Φ001 = 2/1,
+ 02 = 22
(93)
where 0i, 62 are the following 8/22-gradee F-algebra homomorphisms:
Φ : A1 A A22
d1 a1
(94)
Φ : A₂ + A A2,
02 1802,
(95)
which satisfy for all homogeneous elements E A₁ and 02 E A2:
(a1) (2((2)) = (-1)pr((n)) (2(02) (11)))
(96)
Furthermore, A A2 together with 01, 62 is uniquely characterized as a 222222--erdddd R-algebra by the above property (when considering all possible such B and v/,, 4/2).
Proposition D.14. Let (V. q) be finite dimensional quadratic vector space over I, char (har(F) +
with orthogonal sum decomposition:
(V,q) = (V,, @ (V2,
(97)
Then the inclusion maps: CI(V, q.) - C1(V.q) induce an isomorphism of Z/ F F-algebras:
CI(V, q) CI(VV, 91 CI(V2. 92).
(98)
In particular:
dim CI(V, q) = dim CI(VI, 91) dim CI(V2, 92).
(99)
Proof. First consider the canonical inclusion maps, / 1,2:
Φ CI(V, q:) + CI(V,q).
(100)
These satisfy for all (parity) homogeneous elements x; E CI(VI. q1), 1= 1, 2. the condition:
Φ (z1))ZZZZ)) = 21 (-1)prt(e))<pr(e)) 22 = (-1 (-1)prt(s:) pett((z)) (2(22)
(101)
Indeed, we can by linearity reduce to the case that 2 and æ2 are products of vectors V1 V1 and U/ V/2 resp. Since V1 and V/2 are orthogonal to each other by assumption, for those elements we get by the identities D.3
01 0221 + b(1, 22)
12
(102)
This shows the condition above for ii and 22-
We then define the F-bilinear map:
CI(Vi,q1) x CI(V2, 92) - CI(V,q),
(x1, + 11
(103)
which thus factorizes through the F-linear map:
Φ CI(V1, 91) 2 CI(V2, q2) - CI(V,q),
2 0x2 + 21
(104)
We now show that é also respects multiplication. For this let 21,91 CI(Vi, q1) and 22, 1/2
CI(V (22 92) (parity) homogeneous elements. We then get:
/ (12))))) (v1882)
(105)
( (I1))tt(es))ptttt)) (1191)
(106)
(-1)mrt(o)) 21y122222
(107)
(108)
= (12)) (())))))
(109)
This shows the multiplicativity of Φ on homogeneous elements. The general case follows by the F-linearity of é. Note that é also respects the parity grading.
30

---

Now consider the 2/22-graded F-algebra CI(V 91)@ CI(V2, 92) and 2/22- graded F-algebra ho- momorphisms:
CIV2 1 CI(Vi.q:) CI(VI, ( 42),
31 + x 01
(110)
42 CI 92) - CI(VI. 9:)@ C1(V2. 92).
22 + 1822.
(111)
Note that for all homogeneous elements 21 E CI 91), / = 1, we have: 2 (21)) (18x2) (z1
(112)
(- (-1)pr((r))pptaat 2 002
(113)
= (- (-1)pr((1)pp((x)) (2) (1) (18z))
(114)
(- (-1)pr((s)pp(ee)) (21) 122(22)
(115)
We then define the IF-linear map:
0:: V V + 1/2 + CI(Vi, CI(V2, 92), e= (1( (12)
(116)
Note that we have:
x)))² (11( + (22)) (v) + (22(2))
(117)
(v1 + (h( 22((22) + 12((22) - 8hi (v1)
(118)
(11)) + 02(222 + (11) (22)2)) + (-1))an((n)pprrr))))yy)))) (11) (22(22)
(119)
(b(( + 22(33)
(120)
=q1(t) vi (I + (2(2)) 22 (((()ci)))))))
(121)
-q1(t) c) c)(V).9) + 92(22) cu( cI((2.42)
(122)
(q1(01) (22) 4u CI(V2.92)
(123)
q(v) cci(vi CI(())92)'
(124)
By the universal property of the Clifford algebra v uniquely extends to an FFaalebra homomorphism:
v:: CI(V q) CI(V,, & CI(V2,92),
(125)
0ig ................................ 1s. + (t/ (v),,)) + (v(,,)))) ... (111 (vig.1) + (vi..))..)
(126)
One can see from this, by explicit calculation, that also respects the Z/2Z graddgg..ggg..... Furthermore, see that s o 61 th for [ 1, 2, and, also, Ø 2/ di for | = 1,2.
One easily sees that á and s are inverse to each other and the claim follows.
Theorem D.15 (The dimension of the Clifford algebra). Let (V, q) be a finite dimensional quadratic vector space over F, char(E) + 2, n == dim V < cc. Then we have:
dim CI(V,q) = 2".
(127)
Proof. Let 61, .1...., en be an orthogonal basis of (V.q) and V == span(e)) S V. q: 4/ vg- Then we get the orthogonal sum decomposition:
(V.q) (V1, 91) .... + (V., 9n),
(128)
and thus by Proposition D.14 the isomorphism of 2/22-graade algebras:
CI(V.q) CI(V1, (1)..11)8 @ & CI((V.
(129)
and thus:
dim CI(V,q) = dim CIVVV q:) .. dim CI(V,, 9x),
(130)
Since dim Vi2 = and thus:
dim CI(V., = dim (IF @ V2) = 2,
(131)
for = 1,.., we get the claim:
dim CI(V,q) = 2"
(132)
31

---

Corollary D.16 (Bases for the Clifford algebra), Let (V.q) be a finite dimensional quadratic vector space over char(F) + 2, n dim V oc. Let 61, ................................ En be any basis of V. then (eg) basis for CI(V, q), where we put for subser A [n] == { n.::
CA
II
Ce:
cØ Ic((V.))
(133)
VEA
where the product is taken in increasing order of the indices i E A.
Proof. Since CI(V, q) = T(VV) [qq) and
T(V) span {es, @ ... Cim m 2 0, E [m],j E [m]]
(134)
we see that:
CI(V, q) span {ei, Cim
Cim m 2, E [m]. j [m]]
(135)
By several applications of the fundamental identities
D.3
Ci2 Etz E4 6/2 2b(e).:
c/2 C/2 q(er))
(136)
we can turn products Ci2 Ci.. into sums of smaller products if some of the occuring indices agree, say & ig. Furthermore, we can also use those identities to turn the indices in increasing order. This then shows:
CI(V, q) = span {ea | S [n]]
(137)
Since # {AS [n]] 2n and dim CI(V, q) 2" by Theorem D.15 we see that {eA A S [n]] must already be a basis for CI(V.q).
D.5 Extending the Quadratic Form to the Clifford Algebra
We provide the extension of the quadratic form from the vector space to the Clifford algebra, which will lead the constrution of orthogonal basis of the Clifford algebra.
Definition D.17 (The opposite algebra of an algebra). Let (A,+,)) +,.) be an algebra. The opposite algebra (AOP, ..)) is defined to consist of the same underlying vector space (AOp, +)= (A,, +), but where the multiplication is reversed in comparison to A, i.e. for x, E APD we have:
r+y= y-
(138)
Note that this really turns 14 into an algebra.
Definition D.18 (The main anti- intiiinvolution of the Clifford algebra). Consider the following linear map:
H U, which also satisfies u = vu = q(v). By the universal property of the Clifford algebra we get unique extension to an algebra homomorphism:
:: V - CP,,
(139)
B: CI(V,9) + CI(V.g)OP.
Σ
0111
Ui,k;
(140)
VEl
+
Σ
G
U1.1
Ui
(141)
iEl
+
Σ
(142)
YE1
for any finite sum representation with U.y E V and c E F. We call / the main anti- intiiinvolution of CI(V.q).
Definition D.19 (The combined anti-involution of the Clifford algebra). The combined anti-involution or Clifford conjugation of CI(V, q) is defined be the F-aaleebaa homomorphism:
7: CI(V. q) + CI(V.g)09
7xx) = 3(a(z)).
(143)
More explicitely, it is given by the formula:
co
Σ
c
U111
Viiks = + Σ (-1) Ui,ki 2.11
(144)
SET
iE!
for any finite representation with vaj E V and e I..
32

---

Remark D.20. ff...... E V is an orthogonal basis for (V, Then we have for A S [n]:
a(ea) = (-1)lAlea Blea) (-1)( 7(ea)= (-1))=(-)) )e
(145)
Recall the definition of trace linear map:
Definition D.21 (The trace of an endomorphism). Let be vector space over a field of dimension dim y = m < 00 and ④ y + Y vector space endomorphisn [ Let B = (bi...., ..n) be basis for D and B* [bi..... b be the corresponding dual basis of y", defined via: b7(b:) = 8í.j.
Let (ai.) (llg)tl...... be the matrix representation of 4 w.r.t. B: =
Vj E [m].
L(b,) =
aiybs.
(146)
Then the trace of t is defined via:
Tr( _ O3 Σ (7(6)) E.F
(147)
It is well known fact that TD( is not dependent the initial choice of the basis B. Furthermore, Tr a well-defined F-linear map (homomorphism of vector spaces):
Tr: End - 1,,
4 + Tr(2).
(148)
We now want to define the projection of æ E CI(V,,) onto its zero component (0) IF a basis independent way. Definition D.22 (The projection onto the component). We define the F-linear map: ((x) = 2- 2- Tr(x),
<: CI(V,q) +
(149)
where n == dim V and Tr(x) = Tr(L,), where Lg is the endomorphism of CI(V, q) given by left multiplication with æ:
La CI(V,q) CI(V,q),
y y Ln(y) xy.
(150)
We call Ç the projection onto the zero component. We also often write for x E CI(V,q):
00) = ((z).
(151)
The name is justified by following property:
Lemma D.23. Let e1...., en be a fixed orthogonal basis of qV, Then we know that (e (eA)AgIn)
basis for CI(V,q). So can write every CI(V,a) q) as:
I
Σ FA CA.
(152)
AC
with FA E F, AC [n]. The claim is now that we have:
((x) xz-
(153)
Proof. By the linearity of the trace only need to investigate Tr(ea) for C [n]. For A 0, we have eg and we get:
Tr(1) = Σ eg(1 = Σ 1-
(154)
BCIn]
BCIn)
A map from mathematical object space to itself.
33

---

which shows: ((1) = 2 Tr(1) = 1. Now, consider A S [n]] with A * 0. Let A denote the
symmetric difference of two sets. Further, we can write to refrain from distinguishing between signs, which will not affect the result.
Tr(ea) E
(B(Ce)))
(155)
BC
Σ
+
II
((e)) eB(eAA))
(156)
iEAnB
Σ
II
((e))
ÓBBBB
(157)
Σ
II ((e)) 60.4
(158)
BC
SEAnB
0,
(159)
where the third equality follows from the fact that B AAB holds if and only if A = regardless of B. However, A = was ruled out by assumption, so then in the last equality we always have 00,1 So for A + ( have: ((ea) 2 (T(eA) O. Altogether we get:
((eA) = 50.00 = [1, = 0,
(160)
0, else.
With this and linearity we get:
((x) <
Σ .AA
=
Σ
TA ((EA) =
Σ
IA 6 = Ip.
(161)
AC(N)
AC(M]
ACIn]
This shows the claim.
Definition D.24 (The bilinear form on the Clifford algebra). For our quadratic L-vector space (V, q) with corresponding bilinear form b and corresponding Clifford algebra CI(V, q) we now define the following F-bilinear form on CI(V, q):
b: CI(V,)) x CI(V.4) .q + F,
6(x, v) = ((3(x)))
(162)
We also define the corresponding quadratic form CI(V, q) via:
q: CI(V.q) -F
q(x) = =(z,))
(163)
We will see below that and q will agree with and q, resp., when they are restricted to V. From that point will denote b just by b, and with q, resp., without (much) ambiguity.
Lemma D.25. For v, E V we have:
b(u, w) = b(,, w).
(164)
Proof. We pick orthogonal basis e,,...,,, en for V and write:
Dg Cs:
Cj Cg-
(165)
34

---

We then get by linearity:
b(e, w) =
a - b(e,,)) ej)
(166)
0:0y (((()s))))
(167)
a:cy ((ese))
(168)
Σ if
a.ey
((ere)) Σ
0jc; ((ese))
(169)
=0
Σ a,c; (e:) ((1)
(170)
a(ec).
0:Cz ((e
(171)
i=1
n
6 Σ aj Cit Σ
cj cj
(172)
j=1
b(0, w).
(173)
This shows the claim.
Theorem D.26. Let en, be an orthogonal basis for (V,q) q) then (EA)Agia) is an orthogonal basis for CI(V, q) w.r.f. the induced bilinear form b. Furthermore, for x. E CI(V, q) of the form:
x
AA -EA
DA EAA
(174)
AC
ACIn]
with TAAMA E F we get:
b(I,y) = Σ
DA DA
II((et),
q(0) = Σ 23 IIq(e)..
(175)
AC(m]
EEA
ACIn]
EEA
Note that: q(ea) = =11) = 1.
Proof. We already know that (CA)AC(n) is a basis for CI(V q). So we only need to check the
orthogonality condition. First note that for ec = C2, ... Gi we get:
q(ec) = ((3(ec)ec) = C(ei, e, c, ... e,,) = q(ei)) ... q(e.)).
(176)
Now let A, B C [n] with A B, i.c. with AAB A 0. We then get:
b(CA,CB) = ((B(e))e))= = II ((c)) ((CAAB)= =0.
(177)
AEAnB
35

---

This shows that (EA) Ag(n) is an orthogonal basis for CI(V.q). For E CI(V.g) q) from above we get:
b( u) =
Σ
CA'CA.
DA
Σ UB CB
(178)
AÇ
BC ml
Σ
E
IA yB b(EA,EB)
(179)
ACI BC m]
= Σ
Σ
IA YB ((ea) 6A.B
(180)
AçIn B n]]
Σ
DA VA ((AA)
(181)
AÇIn]
Σ ACh
II ((e)..
(182)
:EA
This shows the claim.
D.6 The Multivector Grading
Now that have orthogonal basis for the algebra, show that the Clifford algebra allows vector space grading that independent of the chosen orthogonal basis.
Let (V. q) be quadratic space over a field F with char(F) * 2 and dim V < 00.
In the following we present technical proof that works for fields FF with char(F) (F) + 2. An alternative, simpler and more structured proof, but for the more restrictive case of char( = O, can be found in Theorem D.37
Theorem D.27 (The multivector grading of the Clifford algebra). Let ei....,e, be an orthogonal basis of (V.q) 9). Then for every m ...... n we define the following sub-vector space of CI(V q):
(1(m) (V. q) = span {ei, Cim S
im S n}
(183)
span (Ea A C [m], |A| = m)
(184)
where (1 (V, =
Then the sub-vector spaces (1()) (V, 9). m O,... are independent of the choice of the orthogo-
nal basis, i.e. if bi....,bb ba is another orthogonal basis of (V, 9). then:
(m q) span {bi, ...bi.. S in <... <i m).
(185)
Proof. First note that by the orthogonality and the fundamental relation of the Clifford algebra we have for all + j:
ejej -Ejeis
b,b; -b;bi.
(186)
We now abbreviate:
B(m) = span (bj, ... bim 11s j1 < < fm S kn)
(187)
and note that:
(m)) (V.q) span {ei ... S i1 < .... <...< im S n}
(188)
span {ei, eim 1 ij,..., im S n, Vs + t. i. + ir).
(189)
We want show that for < j1 < < Jm S n we have that:
bj by... E (1) (V.
(190)
Since we have two bases we can write an orthogonal change of basis:
by aiee E V = 110 (V.q).
(191)
36

---

Using this, we can now write the above product as the sum of two terms:
bj ... bj
Σ
aint ... Aimit Cbn ... Ci
(192)
Σ
Oiidi Oim Ei .m +
di
Asmim -ig Cim
Baght.
(193)
Our claim is equivalent to the vanishing of the second term. Note that the above equation for by
already shows claim for m 1. The case m is trivial.
We now prove the claim for m = 2 by hand before doing induction after. Recall that j1 # j2:
b22 = Σ
Oi 0ig
Cig Gig
Oi Gi,jo 6:6
(194)
11 11
Σ
Oi 0iggo
-ii Gig
Σ aiih Gi.jo ((e))
(195)
$17212
= bic
Σ
0i diggd
Cig Big
b(bjr biz)
(196)
61.62
Σ
Gig Ei2
(197)
$1.62 f172
E C1 (V,q).
(198)
This shows the claim for m = 2. By way of induction now assume that we have shown the claim until some > 2, i.c. we have:
bj ... bjm
Biich Oim Ci Bip, E (1) (V,
(199)
S1yoned Vkgll. in
Now consider another by with j == jm+11 jı, k = 1,..., m. We then get:
37

---

by ........ b..
Σ
Biigh aim Ci. Cim
Σ Caige
(200)
Yk Wkoll igghi
Oirhi ... aim.jm OijCi, ... Cin Gi
(201)
Σ
Oirh Oim.fm CijjCi Cim Ci
(202)
Wkell, (i if (m)
Σ
Bighi Om Ci,iCig Bi, Ci
(203)
Hamdon Vieel infil sE(bi...imm)
Σ
aii-ji Oim. Ammt Cij ... Cim Cint1
(204)
Sooddmaoo Vkyl. ix
ΣΣ
Σ
Oigch Oimesin CigiCi Bi Ei.
(205)
1=1
VkFl 1e7 4,
38

---

We have to show that the last term vanishes, The last term can be written as:
Σ
Jh im Oi Ci Cim Cj
(206)
Ohih ... air.e ... mmmmm 0i, Cs2 1, em Cj
(207)
Gi Gi,Jo mmmmm Oi, 63 ". "6 .. Cn Ci, (-1)m-
(208)
frmmi Vesll. 127 =1
Σ
Gi di,.e Asmion Ois Ci "... (-11m==>q(s)
(209)
Σ
Bn 0i, ... mm Oi. + ... BK ... Bim (-11)m==q(e
(210)
Veel..
Oiijj Oi
am Oi.,j Ei '. 966 eim (-1)m-=q(es,)
(211)
ioodd Weyll. f
Σ
Σ
Dishi ... Di... ... aimjjm Diy. Gig . "'.6 ... Pim (-1)m==q(es
1=1
Vkyl. 18 heliimseemmm
(212)
Σ
Σ
anhh ... ai,d. ... ai Ui Big ... 96 Bsn (-1)mm=sq(e,, ((e;)) (213)
bionhh Connhinnto 1==1
Vkol. 1871
Σ
Σ
Σ
airdi ai,ds .0 Agd Ci ".. 6i (-1)m-saa {(ei,)
Mamn Sm is Merii VefLieh
(214)
Σ
Oighh Dirth aimijm Cij "Br." eim (-1)m--
aiaai Gi. g(ei,)
(215)
Σ
Σ
ahija as,gj ... a Im aiat C42 ".. 6im (-1)mm-sq q(e,t)
ii Youndd ((fts.)ooonm))) Wkofl fefl
(216)
Σ
Σ
Girch Oiga ... Oimiim Gig Bi ".. Sr Cim (-1)m->q(ee)
Kanndnn K6..mm Vkzli in
(217)
m ΣΣ
Σ
Circh ... ai.ie ammim Olod C4, .. ".K.. ....... 6im (011)==s((e)))
(218)
39

---

Note that the positional index can occure before or after the positional index Depending of its position the elements e, will appear before or after the element e₁ the product. We look both cases separately and suppress the dots in between for readability.
First consider >
m
1R
Σ
Oiyhh Gieda Oisije" Om auij Cin Sr Ei cim (-1)m==q(es) (q(e.) (219)
Sangend wonddeamm Viefl.
m m
Σ
On Oi.d. Oiyde" Om Di, (-1)(-2ed
ei Bij BB 2x lim (-1))===
(((s))
(220)
Di..j Diece
(221)
1=
with (1)(1): =
Σ
asihh Dbrrt Dirite am Cb2 9K E4. c.m
(222)
Veel. #i
Vk. 64
m aiss ais. aiij q(ei) .e Taal1)
(223)
1=1
173
X72
((,
(224)
with v(s,t)= - 1ymttetaas ais, ass. ais g(ex) e, (1).
(225)
It is important to note that for all 8 * f we have:
y( = ytt, s).
(226)
40

---

Now consider t < 8:
m
12
Σ
abidi Oigde Oseda Jm Did Ci C4 "Ks (-1)m- (((i.) (227)
m 178
Σ
Osi.ji Oieije Oieys Oi Oie ( (-1) Ci. Gi 96. 90- (-1)m==aq q((=)
Sooto Wiel 1121
(228)
Dicd. Bieci
(229)
+ ais. asj. aij ((s))= e. (1)
(230)
m m
t(s,t)
(231)
st,
(232)
[=1 S<t
yss t)
(233)
t=1
s<f
m
m
y(8
(234)
In total we see that both terms appear with different sign and thus cancel out. This shows the claim.
L
Corollary D.28. Let de.ee..... en and bi...., bn be two orthogonal bases (V, q) with basis transition matrix C Jel
Vj E [n].
bj
Σ
Cij
(235)
sE(m]
Then C is invertible and we have the following matrix relations:
CT
((en))C.
(236)
Furthermore, for every subset C [n]] we have the formula:
b =
Σ
det Cr.JJ eIs
(237)
IC
with the submatrix: Cr,J (cus)iErJes.
41

---

Proof. For j,l E [n]] we have:
diag(g(bi); ... .(ba))3 q(b}) 0z
(238)
= =b(b,,b1 bi)
(239)
a
au Cppl b(es,
(240)
G Ck. q(e)) di,
(241)
= (CT diag(q(ei),... q(en)))) /1"
(242)
This shows the matrix identity:
diag(g(bz),.... (b,,) )) c 9((.)))
(243)
Furthermore, we have the following identites:
by = bj. ... ......
(244)
Σ E|m]
Cish
Ci ...
Σ imElul
Gimim Cim
(245)
(Chi .. Cmmmm) (ess ... Emm) ..mmmm)
(246)
in E (oo
D227
Σ
(Gh Cimim) (eis Cim
(247)
h E(n] (1
sgn( , im) (Grih Comum) efi
(248)
( E|n] [01.....m =
Σ
Σ sgn(o) Grayj ... "nromddm Ettemmmm
(249)
i E(n] i
det C1,J
(250)
MC
This shows the claim.
Notation D.29. For m & (0...., n. it is sometimes convenient to put:
(1 (V.q) := 0.
(251)
Corollary D.30. We have the following orthogonal decomposition (w.r.. q) of the Clifford
algebra q) into its F-vector spaces of multivector components:
CI(V, θ (1())) (V,4). m=
(252)
which is independent of the choice of orthogonal basis of (V. q). Also note that for all O,..., n.
dim (m) (V,4)
(253)
m
Definition D.31. We call an element x (1)) (V.q) an m-multiivctor or an element of CI(V, q) of pure grade m. For x E CI(V, q) we have a decomposition:
I = x (0)
+ (1) + (1)
(254)
with (m)) e (V, O,..., n. We call ( the grade-m of z.
42

---

Remark D.32. Note that the multivector grading of CI(V, q) is only grading of F-vecoor spaces, but not of F- F-lleebras. The reason is that multiplication can make the grade drop. For instance. for v E = C1 (V, q) we have ve = q E C1 (V,q), q), while a grading for algebras would require that ve E (1 (V, q), which is here not the case.
Remark D.33 (Parity grading and multivector filtation in terms of multiveetto grading).
1. We clearly have:
0 (V.9) =
(V.q) (v.9)
(1 (V.))
(255)
17==,..... 372 even
m odd
2. It is also clear that for general m we have:
m C) q)
()(mm) (V.q)
CI(V.4).
(256)
simpler proof Theorem D.27 can be obtained if we assume that char(F) = We would then argue follows.
Definition D.34 (Antisymmeetrization)) For m E N and 21...., Im E CI(V,q) q) we define their antisymmetization as:
[x::... 3mm
Σ sgn(o) To(1) ... T.(mm) mn! Em
(257)
where Sm denotes the group of all permuations of [m]. Note that, due to the division by ml need that char(F) = 0 if we want to accommodate arbitrary m E N.
Lemma D.35. Let . be an orthogonal basis of (V. q) and xı, 2i,...,Zm Im E CI(V, g). Then we have:
1. [.13...: 2m] is linear in each of its arguments (if the other arguments are fixed).
2. [1ii (TS ... 5 2T mm - [111 [[rii...... 557 ;aii Tml
3. [m1]... 285 ........0 4rs [m]] = 0 if IR = 21.
4. [e,;.... eim. C₁2 Cin.. if
im = m (i.e. if all indices are different).
Definition D.36 (Multivector grading alternative, basis independent definition). For m ....., we (re- )define:
()(m) (V,4) := span {luis...; venl U1,..., I'm V).
(258)
Theorem D.37. Let ....... On be orthogonal basis of (V, q), char(F) 0. Then for every m O,.... n we have the equality:
(m)) (V. span (EAl ICAIAA A [n],1A|= = mm]
(259)
Note that the rhs is seemingly dependent of the choice of the orthogonal basis while the lhs is defined basis independent way.
Proof. We can write every 0g V as a linear combination of its basis vectors:
Uk
Σ
Ckija Cja-
(260)
43

---

With this we get:
[01i...; ;uml
Σ
II
Chisk legs.... :5ml
(261)
Jm E(m) ke(m)
II J kE(m)
E
Ck.ju lejii...
(262)
Mammmm mm
E
Chiik E/1 ... Gjm
(263)
Aunde Epn kE(m) (m =m
Σ Jm 10 =m
Ckije ((anmmm) kE(m]
II
(264)
E span EEr A [n], |A| = =m}.
(265)
This shows the inclusion:
CI (m) (m
(V.q) S span {EA A [n], |A| m}.
(266)
The reverse inclusion is also clear
CΛ (c. (1 (V. q),
(267)
where A fi and |A| m. This shows the equality of both sets.
D.7 The Radical Subalgebra of the Clifford Algebra
Again, (V, q) be quadratic vector space of finite dimensions dim V n < oo over a field F of char(F) * 2. Let b the corresponding bilinear form of
Notation D.38. We denote the group of the invertible elements of CI(V, q):
(V.4) = {z CI(V, q) By E CI(V, ry yr 1}.
(268)
Let R C V be V's radical subspace. Recall:
R = (ʃ V E o))
(269)
Definition D.39 (The radical subalgebra). We define the radical subalgebra of CI(V, to be:
( == span f1. fu k E No. fi R11= .}} CI(V
(270)
Note that 9/e 0 and that A(R) coincides with cl)
Notation D.40. We make the following further abbreviations:
A M (R) = A(R) C1 (V.
(271)
(1)
(R) = span (ʃ1 ...... kalk2 1,21,1 E k).
(272)
A (R) = Fx (21) (R),
(273)
[X]
(R) = Fx span {fi fu k even, fi RR1=1, 1.....,k
(274)
A (R) = 1 + ^ (21) (R),
(275)
(+] (R) span (h1 ... f& 1k even fi E R,, =
(276)
Here, 1 denotes the set of invertible elements of F.
Lemma D.41.
1. For every h A (22) (R) there exists k 2 such that:
A+1 0.
(277)
In particular, no h A (2) is ever invertible.
44

---

2. Every y E (R) - A(R) \ (22) (R) is invertible. Hss inverse is given by:
y = c (1 h h2 --1.(-11)hh))
(278)
where we write: y = c. (1 + h)) with e E I8x h ME (R)), and such that h
3. In particular, we get:
- A
(279)
Proof. Items and are clear. For item we refer to Example E.21
Lemma D.42 (Twisted commutation relationships). the anticommutation relationship:
1. For every f E R and E V we have
fv= 226(f,) -uf.
(280)
2. For every f E R and x E C)(V.q) we get the following twisted commutation relationship:
a(x)f = (x - )f = fxx + x!2 fx.
(281)
3. For every y E ARR and every C1 (V, 9) we get:
xy = yz.
(282)
4. For every y A(R) and V ger:
aly)v = (y) - y = v(yl0) | + y = uy.
(283)
5. For every y A (0) (R) (of even parity) and every x E CI(V, q) we get:
yr xy.
(284)
Remark D.43. A direct consequence from Lemma D.42 that the even parity parts: A [x] (R) and A (+) (R) all lie in the center of CI(V, 9), which we denote by 3(CI(V,g)
(0) (R))
(R) S R S A (R) S 3(CI(V.
(285)
i.e. every y E (0) (R) (of even parity) commutes with every CI(V, q). For more detail we refer
to Theorem D.47
In the following we will study the center of the Clifford algebra 3(CI(V, q)) more carefully.
D.8 The Center of Clifford Algebra
In the following final subsections, we study additional properties of the Clifford algebra that will aid us in studying group representations and actions on the algebra the upcoming sections.
Let (V. q) be quadratic space over a field F with char(F) * 2 and dim V < oc.
Definition D.44 (The center of an algebra). The center of an algebra A is defined to be:
3(A) == {2 E A VI E A. A. = =
(286)
Lemma D.45. Let e an orthogonal basis of (V, q). For A C [n] = (I,... .n. let eA = - A es be the product in CI(V, q) in increasing index order, Ep == 1. Then we get for two subsets A, B C [n]:
CΛCB = (-1)14) B IAn CBEA-
(287)
45

---

In particular, for & A we get:
CAC; (-1)/4) eyeA.
(288)
and:
eyeA -
+ +))
(289)
where is the position of j in the ordered set AÚ {i}.
For i E A we get:
Exc; = ---)((-1))---- BGA = -ee
q
(290)
and:
EYc - Gex= --) + 1){(e))amm)
(291)
where s the position of i in the ordered set A.
Proof. Let B == Li jaBi S [n].
CACB = II
6 II Cj
(292)
YEA
JEB
(-1)LA1-
Il
ej
(293)
EEA
jEB
(-1)|BIAAI-E
1[EAA)
CBCA
(294)
= (-1) BIIA|-IANBI EBEA
(295)
For the other two identities, similarly make of the fundamental Clifford identity.
Lemma D.4.. Let (V, be a quadratic space with dim V < oc. Let eI...., on be an orthogonal basis for (V, 9). For x E CI(V, q) we have the equivalence:
E 3(CI(V, q))
Vi [n].
Ie; = G,T.
(296)
Proof. This is clear as CI(V, q) is generated by exg Cky-
Theorem D.47 (The center of the Clifford algebra). Let (V, q) be a quadratic space with dim V < co, char F + 2, and let R S V be the radical subspace of (V, 9). Then for the center of CI(V, q) we have the following cases:
1. n is odd then:
3(CI(V,4) = ^ 19 (R) a (V.
(297)
2. Ifn is even then:
3(CI(V. A (R).
(298)
In all cases we have:
A A (R) S 3(CI(V
(299)
Proof. Let 61, e....., Pn be an orthogonal basis for (V.9). The statement can then equivalently be expressed as:
46

---

1. If n is odd or 0 (on all vectors), then:
3(CI(V, q)) = span { CA, ((n)
|A| even, Vi E A.q(e) = 0}
(300)
A (0) (R) CI (V,
(301)
2. If n is even and q # (on some vector), then:
3(CI(V q)) span en |A| even, Vi A.q(e:) 0)
(302)
[0] (R).
(303)
Note that if n is even and q 0 then both points would give the same answer, then V R and
thus:
CI (m) (V.q) - -A (n) (R) S
[0]
(R).
(304)
We first only consider basis elements with A S [n)] and check when we have 3(CC(V..))
We now consider three cases:
1. 0. We always have: ey 3(CI(V.g)).
2. [n]. For every i [n] we have:
einje = (-1)"=1 eqenn [e,)) [plM)
(305)
So [|j] E ((CI(V iff either n odd or = (on ez).
3. + $ [n]. Note that for i A we have:
CΛc (- (-1) GRA = +e(e)) eA(())
(306)
while for j & A we have:
excy = (-1)AA1 eyeA.
(307)
The latter implies that for e 3(CI(V, to hold, |A necessarily needs to be even. Since in that case (-1) 11 we necessarily need by the former case that for every i E A, (ee)) = O.
Now consider a linear combination x ΣAs( CA ee E CI(V,9) with CA E F. Then abbreviate:
Z span {eA |A S [n], 3(CI(V,g)) 3(CI((V))))
(308)
We need to show that æ E Z. We thus write:
x = 4 2,
= Σ
CA
CA.
2:= Σ CA E
(309)
Ag
Ag
eA
CAEZ
So with x, z E 3(CI(V, q)) also y E (CC q)) and we are left to show that y = 0.
First note that, since ey E 2, there is no co-component in
We now have for every [n]:
0 = ye,-
CiV
(310)
= Σ CA (Eyes - -(EA)
(311)
nÇin)
2
Σ
CA (Eyes cAAe, -GGea) + Σ
CA (e
- (exe--eee -GEA)
(312)
AC m
AC
EA#2
cA42 i
iEA
=
Σ #((-1)1A1 +)q(Cs)cC eA\() + Σ +((-1)1A1+1 + +1CA CAU
(313)
AC
e
ig
47

---

Note that for B, S [n] with A * B we always have:
A (1) # B (i): A\ { {i},
if A,i E B, if A B, if $ A,i if e A, B.
(314)
(315)
AÙ B\ {i].
(316)
( BÚ {i),
(317)
So the above representation for ye, -ey is already given in basis form. By their linear independence we then get that for every A [n] with a & and every E [n]]:
= ((-1))A) + L)q(e;)cA.
for E A,
(318)
0 ((-1) 1)CA.
for & A.
(319)
First consider the case e[n] E 2. By the previous result we then know that is even and / 0. So there exists e, with ((e,) # 0. So the above condition for E [nx] then reads:
0 = 2q(e))cint
for i E [n]],
(320)
which implies Cjs] = 0 as ((e)) # 0. So does not have a (nj] compoonnt.
Similarly, for S [n] with # [n] and a $ and |A] odd there exists & A and the above condition reads:
2cA,
for & A,
(321)
which implies сд 0. So does not have any ea-components with odd |.]..
Now let A S [n] with A L [n] and ед & and |A| even. Then by our previous analysis we know that there exists i A with ((:a) # O. Otherwise E 2. So the above condition reads:
0 = (e(e))c:
E A,
(322)
which implies сд 0 as ((e)) ) * 0. This shows that y does not have any e comommnnnntt with [A].
Overall, this shows that 0 and thus the claim.
D.9 The Twisted Center of the Clifford Algebra
Notation D.48. Let el,...,en be an orthogonal basis of (V,q). For A C [m] (I.....m} let еΛ == le e; be the product in CI(V, q) in increasing index order, eq == Then aCin) basis for CI(V, q).
Definition D.49 (The twisted center of a 35/2Z-graded algebra). We define the twisted center of Z 2/2Z graded algebra A the following subset:
R(AA = f Ye A. or p + (y) "'))'"'
xy
(323)
Theorem D.50. We have the following identificution of the twisted center with the radical subalgebra of the Clifford algebra CI(V, 9) and the set:
*(CI(V,9)) A(R) ty CI(V. (VEE V. =yy
(324)
Proof. Let y E A(R) then by Lemma D.42 we get:
yr 00 + a())=!"! = 1 + x x. zy.
(325)
This shows that:
y E A(CI(V,))),
(326)
and thus:
A(R) C R(CI(VV)))
(327)
48

---

Note that the following inclusion is clear as V _ c]):
R(CI(V S tu CI(V Vo V. a(y) vy).
(328)
For the final inclusion, let y = Eng(y cp eB E CI(V,q) such that for v E V we have aly) vy. Then for all orthogonal basis vectors 6 we get the requirement:
ezy aly)e,,
(329)
which always holds (e;) = O, and is only a condition for q(e,) + O. For such e, we get:
CB
CB +
Σ
cB
GB
Σ
CB
cB +
Σ
CB CB
(330)
BC
RC
BC
[C
[B] even
[B] even
[B] odd
B odd
ig
c/y
(331)
aty)e
(332)
CB CB
Σ
CB CB
Σ
CB eB +
CB CB C₁
(333)
BC[m]
BCIn]
BC[n]
BC ml
[B] even
[8]] even
[B] odd
[2] odd
<E B
i
igB
Σ BC
CB GB +
Σ
CB BB
Σ
CB GB
CB CB
(334)
BC
BC in]
BC
BB even
[B] even
[8] odd
iE B
B
iE
en en
Σ [C
CB EB
Σ
en en
Σ cn 6B
(335)
BC [B]] even
BC
BC(M]
[B] even i
B odd
[B] odd i
iEB
Since e with q(e;) o invertible, we can cancel e, on both sides and make use of the linear independence of (eB) to get that:
= if i
(336)
Since for given B this be concluded from every e with q(es) * 0 only have en # 0 if ((ez) = for all j E B. Note that elements ej with q(eg) = 0 that are part of an orthogonal basis satisfy ej R. This shows that:
Σ
cB en E span {EA | A V.e A. = A(R)
(337)
BCM
This shows the remaining inclusion:
(v CI(V.)) a(y)v= - C A(R).
(338)
This shows the equality of all three sets.
E The Clifford Group and its Clifford Algebra Representations
We saw that -iieuo-Die (Theorem C.13 generates the orthogonal group of (non-degenerate) quadratic space by composing reflections. Considering this, we seek the following a group representation that acts on the entire Clifford algebra, but reduces to reflection when restricted to V. Further, we ensure that the action an algebra homomorphism and will therefore respect the geometric product.
49

---

E.1
Adjusting the Twisted Conjugation
Recall the notation CI (V.q) := fa e CI(V.q) CI(V, my my y yr=
Motivation E.1 (Generalizing reflection operations). For v, w V with q(w) o the reflection of onto the hyperplane that is normal to to is given by the following formula, which we then simplify:
b(w, v)
Tw (v) = U - 2
b(w, w) w
(339)
www/q(w) 2 b(w, v)
q(w) w
(340)
=- -w(wwr 26( w. u)) (q
(341)
-wow/qqw
(342)
www""
(343)
So we have Tw (v) www- for v, w E V with (w)) + O. We would like to generalize this to an operation P for w from a subgroup of S CI* (V, (as large as possible) onto all elements E CI(V, q). More explicitely, we want for all w1, W2 I and æ1, E CI(V. q). v. w E V, (w) *
p(w)(v) = - wow = rm) (p(wz) o p(wr)) (x1) (w22u)(ar),
(344)
(345)
((r + x22 = p(n))(())) ) + (12)) p(w1) p(w))({z))((t)))((
(346)
(347)
The second condition makes sure that CI(V, q) will be group representation of i.e. get group homomorphism:
p: - Aut(CI(V, 4));
(348)
where Aut (Ct(CI(V,a) denotes the set of automorphisms CI(V, CI(V,q) q). In the literature, the the following versions of reflection operations were studied:
Po(w): x + w
PI(w): x + a(w)x=
p2(w) x + wra(w)-1
(349)
Since po(w) is missing the minus sign il only generalizes compositions of reflections for elements wK. wi E V, a(vi) + 0, of even parity k. The map pi (w). the other hand, takes the minus sign into account and generalizes also to such elements w w W.... of odd party k prt(w) as then: a(w) = (-1)prn((m)a w. However, in contrast to po(w), which is algebra homomorphism for all w E C1* q), the map pi (w) is not multiplicative in as can be seen with U1 02 E V and w E with q(w) #
(1(w)(1122) = a(w)r
(350)
(-1 (-1)pet(w))wreg wor w (-1)ott woy--
(351)
= -(-1)Dr((w), pi (w))
(352)
# (i(w)(o1)p) (w2)
(353)
The lack of multiplicativity means that reflection and taking geometric product does not commute.
To fix this, it makes sense first restrict pr (w) to V, where it coincides with Tw and also still is mul- tiplicative in w, and then study under which conditions on w it extends to an algebra homomorphism CI(V, 9) CI(V,q) q).
More formally, by the universal property of the Clifford algebra, the dbstrruction.fr:
p(w): : V CI(V,q), ((w)(v) == a(w)ow wn(w))ww Ty(w) = n a(w), (354)
with general invertible C1 (V. q). extend to FFaagebra homomorphism:
p(w) CI(V. + CI(V.
(355)
50

---

is the following:
Au V.
q(v) (p(w)(t))²
(356)
((w)(t))(w))v))
(357)
(358)
wn(w)un(w)een
(359)
which reduces to:
VU E V.
q(0) = ((w))n((()))
(360)
The latter is, for instance, satisfied if ((w) commutes with every v V and q(w)². In particular, the above requirement satisfied for all E CI* CI*(V, (V.q) with ry(w) E {+1), which equivalent to
a(w) tw, which means that w is homogeneous element of CI(V, q) in the parity grading. This discussion motivates the following definitions and analysis.
Notation E.2 (The coboundary of a:). The coboundary 1 of a is defined on E ECI* (V, as follows:
7: CI (V,4) + CI (V, q).
<(w) :- w a(w).
(361)
Remark E.3. 1. n is crossed group homomorphism (aka 1-cocycle), i.e. for w1, W2
CI (V, q) we have:
((w)w2)= = w)( m)²
with
n(w)) = W2 10-2m(w (1(w))
(362)
2. For w (V,q) we have:
a(n(w)) (awwwww'uu n(w)-
(363)
3. For C1* (V.q) we have that is homogeneous element in CI(V, q), in the sense of parity, if and only if n(w) E {+1}.
Definition E.4 (The group of homogeneous invertible elements). With the introduced notation we can define the group of all invertible elements of CI(V, q) that are also homogeneous (in the sense of parity) as:
( == [C1* (V, (v u (v.g)aaaa))))
(364)
[w e C1 (V, n(w) [=1)}.
(365)
Notation E.5 (The main involution revisited). We now make the the following abbreviations:
a = id: CI(V, q) CI(V,q), a= = a CI(V, q) - CI(V,q),
a = -!! + [1' x.,
(366)
a (z) = _.! 11
(367)
For E CI(V, q) then have;
apr((w): CI(V. + CI(V,q),
Prt(i)) (x) = [0 + (- (-1)prt(")
(368)
Note that ppr((w) is FF-aleebra involution of CI(V, that preserves the parity grading of CI(V, q).
We also need the following slight variation a"", which many, but not all cases, coincides with prtw)) For w C1* (V. q) we define the w-twisted map:
CI(V. + C)...)). as
("(x) == I! + 11
(369)
Remark E.6. Note that for (g(x((V,q) have that r)(w) (pjoot(())))) and thus a pet((w)) in which case " is an F-algebra involution of CI(V, q) that preserves the parity grading of CI(V q).
Definition E.7 (Adjusted twisted conjugation). For w E C1* (V, q) we define the twisted conjugation:
p(w): CI(V.q) + CI(V.g),
p(w)(x) == wz 10 -1 + aww))=""
(370)
w (at0) + n((warl!)) w-
(371)
wa ana"(an (x)w=
(372)
51

---

We now want to re- re-investigate the action of p on CI(V, 9).
Lemma E.8. For every w E C1(x| (V, q) the map:
p(w): CI(V,q) - CI(V, 9). x p(w)(z) w (x²)) | + n(w))zll)) w w-
(373)
an F-algebra automorphism that preserves the parity grading of CI(V q). Its inverse it given by p(w-))
q). Since ppe((w) is an F-algebra automorphism that preserves the parity grading of CI(V, q) so is " for w E Ci (V,
Proof. First note that a agrees with the involution (pe((w) for E (V,
Furthermore, the conjugation po(w): x + wow is an F-allgera automorphism, which preserves the parity grading of CI(V, q) if w E (V(*) q). To see this, note that 0 = prt(1) prt(ww prt(w) () + prt(w 1)). As such, prt(w) prt(w Then, prt(wrw 1) prt(w) + prt(z) prt(w (w 1 2 prt(w) + prt(æ) = prt(æ). Here, we use the fact that the Clifford algebra is 2/22-- graded.
So, their composition p(w) = (o(w) o a" is also an F-algebra automorphism that preserves the parity grading CI(Va) for w E ci (V, q).
As direct corollary we get:
Corollary E.9. Let F(To........ Ti,e) F(To, be polynomial in 28 variables with coefficients in I.. Let II...., It E CI(V, q) be elements of the Clifford algebra and w E Ci (V, (V,q) q) be a homogeneous invertible element of CI(V, q). Then we have the following equivariance property:
p(w) (F(ZT)
10]
(1)) = (()( ..... p(1)) (()))))
(ox)
(374)
Proof. This directly follows from Lemma E.8
Futhermore, we get the following result:
Theorem E.10. The map:
p: ClIx1 (V.4) - Aur (CI(V.))
w p(w),
(375)
well-defined group homomorphism from the group of all homogeneous invertible elements of CI(V, q) to the group of F Flalgebra automorphisms of CI(V,q) that preserve the parity grading of CI(V.q) q). In particular, CI(V, q). c1 (V,q). (V q) group representations of Ci1x)
(V.x)VV (V.q) q) via
Proof. By the previous Lemma E.8 we already know that is well-defined map. We only need to check if is group homomorphism. Let wi, wa E C)I*I(V, q) and CI(V, q), then we get:
(p(w2) o p(w1)) (x) p(w2) (p(z)(((()(z)))
(376)
p(u2) (wiapa(w.)) (z)
(377)
prt(w) ( (a (x) w.
(378)
uya (w) (o (2) (w)) -
(379)
wzz( (r(( pr wy prt(ws)) (1)ttt)) prt(
(380)
wywwa (2 (x)),i'w5 w
(381)
(wzmr) ) pri(waw)) (w)
(382)
(())
(383)
52

---

where we used the multiplicativity of (p (z):
a" (x)a² (v) = ((-1)orc(w), الl + xx 2101) ((-1) y + y/0)
(384)
(-1)pr((w)) ( + x 1011 (00)
+ x (211 + x 10 00
(385)
= a ((z))"1 + (zz)
(386)
(xy).
(387)
This implies:
p(w2) o p(ur)) = p(w2wr1),
(388)
which shows the claim.
Finally, we want to re-check that our newly defined p. despite its different appearance, still has the proper interpretation of a reflection.
Remark E.11. Let w, u E V with q(w) + O. Then (v) is the reflection of o w.r.f. the hyperplane that is normal to w:
((w)(o) wa" (o)w"
-wow- (v))
(389)
Remark E.12. The presented results in this subsection can be slightly generalized as follows. If
W C1 (V.g) such that n(w) E A(R) then by Theorem D.50 we get for all E V:
wn(w) a(n(w)))= n(w) b.
(390)
This implies that for all u E V.:
(w) = (n()n((())))))))) qu))
(391)
and thus for all u E V. By the universal property of the Clifford algebra the map:
(a(w)ow ) (a(w)) (aa()) ow ow = q(v).
(392)
p(w): V + CI(V, a),
+ a(w)rw
(393)
uniquely extends to FF-algebra homomorphism: p(w): CI(V,q) CI(V.q).
(394)
with:
I == +
Σ
U1,1 ... U.kk.
(395)
iE!
+ 00 + Σ
C a(w)viiiw a(w).u, w
(396)
iEl
co +
( ... ..nn(uu w-
(397)
iEl
Co
4 U2
Ug + n(w)
Σ
C: U1,1
(398)
SET ky even
sEl
W. [0 + n(w) 1l)
w
(399)
To further ensure that the elements w E CI* (V,q) with the above property form a group we might need further restrict to require that ry(w) E A (R)) At least in this case we get for w1, W2 E C1* (V, q) with n(w1), E A xx] (R) S 3(CI(V. q)). see Remark D.43 that:
((w2w))= = wi ((ww))in(w1)> = wI " (nn((22)(()))) = ([(2))( E A (R). So the following set:
(400)
C: C:= w E ( (V. n(w)= E A [x] (x))
(401)
is subgroup of C (V, q) where every E C defines an algebra homomorphism:
p(w): (V) + CI(V,q),
p(w)(x) = w (xx²) + n((w)=l") w-1.
(402)
53

---

E.2 The Clifford Group
Motivation E.13. We have seen in the last section that if we choose homogeneous invertible elements w E CV q) then the action p(w), the (adjusted) twisted conjugation, is an algebra automorphism of CI(V, q) that also preserves the parity grading of CI(V, q). in particular, p(w) is linear and multiplicative.
We now want to investigate under which conditions on w the algebra automorphism p(w) also preserves the multivector grading:
CI(V. q) (V.9) @ (V.)) @ t.... e (1(m) (V.q) @ ① (V.)).
(403)
If this was the case then each component (l(m) (V,9) would give rise to a corresponding group representation.
To preserve the multivector grading we at least need that for v V = C (V,q) we have that also (v) E (1 homomorphism, i.e. for w homogeneous and invertible, and such that ((w)(v) E V for all V, we already get the preservation of the whole multivector grading of CI(V,q).
= V. We will see that for w E CI* (V. q) such that p(w) is an algebra
Again, let (V, q) be quadratic space over a field F with char(F) F 2 and dim V n oo and e1..... En and orthogonal basis of (V, q).
Remark E.14. In the the following, we elaborate on the term Clifford group in constrast to previous literature. First, the unconstrained Clifford group is also referred Clifford-Lipschitz group in honor of its creator Rudolf Lipschitz (LS09]. Throughout its inventions, several generalizations and versions have been proposed, often varying in details, leading to slightly non-equivalent definitions. For instance, some authors utilize conjugation as an action, others apply the the twisted conjugation, while yet another group employs the fwisting on the other side of the conjugation operation. Furthermore, some authors require that the elements are homogeneous in the parity grading, where others do not. We settle for a definition that requires homogeneous invertible elements of the Clifford algebra that act via our adjusted twisted conjugation such that elements from the vector space V land also in V. The reason for our definition is that we want the adjusted twisted conjugation to act on the whole Clifford algebra CI(V, q). not just on the vector space V. F Furthermore, we want it to respect, besides the vector space structure of CI(V, q), also the product structure, leading to algebra homomorphisms. In addition, we also want that the action to respect the extended bilinear form b, the orthogonal structure, and the multivector grading. These properties might not (all) be ensured in other definitions with different nuances.
the Lipschitz group or
Also note that the name Clifford group might be confused with different groups with the same name in other literature, e.g., with the group of unitary matrices that normalize the Pauli group or with the finite group inside the Clifford algebra that is generated by an orthogonal basis via the geometric product.
Definition E.15.
1. We denote the unconstrained Clifford group of q)(V,q) as follows:
i(V.q) q) == {u C1 CV*((V.g) Yu e V.p(w)(u) e v}.
(404)
2. We denote the Clifford group of CI(V, q) as follows:
r(V. q) == CI x (V.q)
(405)
{ E CI (V.9) n(w) (+1) A V.p(((()) EV..
(406)
3. We define the special Clifford group as follows: r [0) (V.q) = i'V, q) n (V.) ci (V,q).
(407)
Theorem E.16. For w E (VV.q) q) and x E CI(V, q) we have for all m O...., n:
p(w)(z(mm) (z))
(408)
In particular, for (1(m) (V, q) also have (z)) e () (V, (V,
54

---

Proof. We first claim that for w E ((V.g) the set of elements:
b = (e))()),..... bn == (ew)
(409)
forms orthogonal basis of (V, q). Indeed, since (()) V, by definition of I(V, the orthogonality relation, + j:
0 2b(es, ez) = ezey eyei
(410)
transforms under p(w) to:
(0) plw) (eee+ + efe,)
(411)
p(w)(c))p(u\(e)) (lw(e)))ee b,b, b,bb;
(412)
(413)
2b(b, by).
(414)
This shows that .... b, is an orthogonal system in V. Using p(w=1) we also see that the system linear independent and thus an orthognal basis V. By the masis-independence of the multivector grading Cl(m)(( (V, q), see Theorem D.27 we then get for:
......,,,,
Σ
Ci Eig
Ciem (m( (V,9),
(415)
the relation:
((w)(x)) =
Σ
Ci bi ... b.. E (1)) (V.
(416)
11 fic
This shows the claim.
Corollary E.17. The map:
p: D(V. Alg (CI(V, q)),
p(w),
(417)
is well-defined group homomorphism from the Clifford group to the group of F-algebra automor- phisms of CI(V, q) that preserve the multivector grading of CI(V, q). In particular, CI(V, q) and C1 (V. q) for m = O,...,n, n, are group representations Tf q) via p.
Corollary E.18. Let F(T,.... Te) FT......, TT] be a polynomial in é variables with coefficients in F and let & E (O.....nk nn.. Further. consider & elements II, E CI(V.q). Then for every I(V, q) we get the equivariance property:
p(u) (F(EI..... Fll..w)a........l.))))...
(418)
where the superscript (k) indicates the projection onto the multivector grade-k-part of the whole expression. Example E.19. Let w with qww) + 0, then E gVV.g).
Proof. It is clear that w is homogeneous in the parity grading, (w) -1 For E V we get:
p(w)(v) (w) = = b(v, w)
q(w) w E V.
(419)
This shows w E I(V,q).
Example E.20. Let e, V with b(c, f) = 0 for all V. and put: : 1 ef a (V. (V Then y g),.)).
Proof. First, note that for all v V we get:
uu= = -vf-
(420)
Next, we see that:
-- 1 eff.
(421)
55

---

Indeed, we get:
([+ef)(1-ef) = ltef ef - efef
(422)
=leeffe
(423)
= =1
(424)
=1.
(425)
Now consider & E V. We then have:
(())(1) = 077)0711
(426)
(1 + +ef)r((l -f)
(427)
= (1 + eef( - vef)
(428)
cfe vef efvef
(429)
= v evf vef - eveʃs
(430)
(ev + re) (f) eve
(431)
26(e, v)f
(432)
E V.
(433)
This shows E (VV.)).
Example E.21. Let f fr E V be a basis of the radical subspace R of q). In particular, we have b(u. f,) 0 for all v V. Then we put:
9 1+
with h E Espan' (1r
fx, 1s
even,
Ki < k2 kg sr) S CI(V. q).
(434)
Then we claim that E ((V,q) and p(g) = idcivaa)
Proof. Since we restrict to even products it is clear that g E C1 (V, q). Furthermore, note that, since lives in the radical subalgebra, there exists a number k 2 1 such that:
k+1 = 0.
(435)
Then we get that:
g =1 h 122 + ....
+ (-1)**².
(436)
Indeed, we get:
(1 + h)
Σ
(-1
= -
(-1)%) +
(-1)
(437)
1=0
1=0
1=0
+
1)
(-1) + (-1)* k+1
(438)
= 1.
(439)
Furthermore, g lies in the center 3(CI(V, q)) of CI(V.q) q) by Theorem D.47
Then for v E V we get:
((g)(v) = a(g)vg-
(440)
gog
(441)
vgg
(442)
u
(443)
V.
(444)
So, g E g(( and acts as the identity on V, and thus on CI(V. via
56

---

1. have the following identity for E.3 The Structure of the Clifford Group
We have identified the Clifford group and its action on the algebra. In particular, our adjusted twisted conjugation preserves the parity and multivector grading, and reduces to reflection when restricted V. We now want further investigate how the Clifford and the Clifford groups act via the twisted conjugation p V and CI(V, q). We again denote by R C V the radical subspace of V w.r.t. q. We follow and extend the analysis of [Cru80 Cru90 DKL10
We first investigate the kernel of the twisted action.
Corollary F.22 (The kernel of the twisted conjugation). the twisted conjugation:
ker (plcIx = {w E C1 (V.g) (V idcicv.)) 1 A x (R).
(445)
2. For the twisted conjugation restricted to the unconstrained Clifford, group and to V the kernel is given by:
ker (p: [(V. - ((VV)))
== (w f(V. p(w)lv = -dlll }
= A (R). (446)
3. For the twisted conjugation restricted to the Clifford group, the kernel is given by:
ker (p: I(V,q) + (entAig(C((V,OO)) == {w E r(V. ((w) = idcu(va)))
(447)
[X] (R).
(448)
Proof. This follows directly from the characterizing of the twisted center of CI(V, q)) by Theorem D.50 Note that we have for w E C1 (V:q):
p(w) idcn(v.a) - Vo CI(V. q). p(w)(x) = x
(449)
Vx E CI(V, q). wx w0 -1 + a(w) 2
(450)
Vo CI(V, q). wxx + a(w)a!!! 3W
(451)
w E R(CI(V, A(R).
(452)
From this follows that:
w CI (V, 9) n (R)) A (R).
(453)
The other points follow similarly with Theorem D.50 For the last point also note:
A (R)) = A (R).
r(V, q) = ((V. (V.)),
(454)
This shows the claims.
Lemma E.23. 1. For every w E CI* q) we have:
p(w)(R idR.
(455)
2. For every w E (V,) we have:
p(w) = =id(RR)
(456)
3. For every g A (R) we have:
plg)lv idv.
(457)
4. For every g E E (x((R) we have:
p(g)lcu(V.) idoiva.))
(458)
57

---

Proof. This directly follows from the twisted commutation relationship, see Lemma D.42 For
w E CI (V,9) and f E R _ V we have:
p(w)(1) = a(w) fw= = f'ww -1
(459)
For w E (V. q) the map p(w) is an algebra automorphism of CI(V, q) and satisfies p(w)(f)
for all f E R. So for every y A(R) we have:
EG Ske Sk. f..
((w)(v) p(w)
(460)
Σ
G. (f.) (f.) ((u
(461)
ie!
Σ
Gi ". ft.
(462)
Kel
y-
(463)
For g A (R) and V get:
(())() = alg)vg vgg v.
(464)
For g E A (R) the map p(g) is an algehra automorphism of CI(V, 9) and satisfies p(g)(u) = v for
all E V. As above we see that for x E CI(V, we get:
pl))(z) = p(g)
Σ iE!
Uk
0z
(465)
Σ
p(g)( ... (ol)
(466)
U
(467)
iel
(468)
This shows all the claims.
From Lemma E.23 we see that p(w)lA(R) =idACR) for w E (V q). Together with Corollary
E.18 we arrive slightly more general version that allows one parameterize polynomials not just with coefficients from IF, but also with elements from A (R), and still get the equivariance w.r.t. the Clifford group I(V.)): q):
Te4al be a polynomial in E s variables with coefficients in F and let k E (O....,n), Further, consider i elements I II...., Zy. E CI(V, q) and
Corollary E.24. Let F(Ti....,TTa Te+a) E
elements 91,....36. Us AR).. Then for every w I(V,q) we get the equivariance property:
p(w) (F(r1....., V...,,........ wa)(k)))
(469)
where the superscript (k) indicates the projection onto the multivector grade- k- k-part of the whole expression.
We now investigate the image/range of the twisted conjugation.
Theorem E.25 (The range of the twisted conjugation). The image/range of the Clifford group 1'(V, q)
under the twisted conjugation restricted V coincides with all orthogonal automorphisms of (V, q) that restrict to the identity idg of the radical subspace R C of (V,
ran (p T(V. - GI.(V)) = (v.
(470)
Again, recall that the kernel is given by:
ker (p: r(V.q) + GL(()) = - (M(
(471)
58

---

Proof. We first show that the range of when restricted to V will consistute an orthogonal automor- phism of (V, q). For this let eI....,y, en be an orthogonal basis of (V. q) and R the radical subspace of (V, 9), r = dim R. W.l.o.g. we can assume that 61,-.., Cm, 6m-+ 1, ..... Cmtr with:
R span fem+1....., emtr);
E == span {e1,...; cm)
(472)
For w E r(V,g) q) we now apply p(w) E GL(V) to the basis elements Note that by definition of I(V,9) q) we have (()) E V. With this we get:
2bl(())())) = p(w>(et)p(w)(e))) + (())(s))((((e))
(473)
p(w) (ere teje))
(474)
=((w) e;))
(475)
2z(
(476)
This shows for v =E== a1: Gz
((p())(v)) = q Σ a (())
(477)
n
at (c,(w((,))
(478)
Σ af ((()
(479)
90
a es)
(480)
q(v).
(481)
Since p(w) is also a linear automorphism of V with inverse 1) we see that:
p(w)lv E O(V.)).
(482)
By Lemma E.23 we also see that:
p(w)|R idR.
(483)
Together this shows that:
p(w)lv E (v.(V.)).
(484)
This shows the inclusion:
ran (p: T(V.q) + GL(V)) Cx(
(485)
Recall the definition of the set of radical preserving orthogonal automorphisms:
((E,g)e) 0mxr M(r,
(V. = { E 0(V,4) Φ = =idæ}=
(486)
idR.
(EE, glæ) x m(r.)).
(487)
So an element & O (V, q) can equivalently be written as:
0
0
0
0 I
0
Φ- =
M
1
I
o
M
(488)
( 0
On
9)
1 9)....
T Mm,r (ML,, 9).
M1, 0
(489)
0
where 0 O O2 is a product of k S m reflection matrices O by the Theorem of Cartan- Dieudonné C.13. and M = E E M,, where the matrix Ma.j Ci.j Ii,j only has the entry G.j E FF at (i, (and 0 otherwise). Now let wy V be the normal vector of the reflection O with ((w)) # 0 for I,...,k, and for i = 1,..., m and j = I,..., r put:
Vig == 1 + C.j e.ettj E CV(Vq)).
(490)
59

---

and further:
W == W0 ... W. 11,1 ... mm.r E CI(V,)).
(491)
Note that by Examples 1.19 and E.20 we have:
(w LVViq(w) + 0} _ CV(
(492)
{ Y/ = 1+ + c
E
cisemty Gr EF,i= m. -...... ST(V.)).
(493)
j=1
which implies that w g(V.q). With this we get:
p(w)= p(w1) p(wx) p(mn.)) p(Tmma))
(494)
O1
(o 7 (M..
M1, 9) 1)...
(M..
9) Mm.
(495)
(496)
This thus shows the surjectivity of the map:
p: I(V.9) - Oz (V.
(497)
This shows the claim.
We can summarize our finding in the following statement.
Corollary E.26. We have the short exact sequence:
[x]
1
R
r(V.q) -a (v.q)) +11
(498)
From the above Theorem can now also derive the structure of the elements of the Clifford group- Corollary E.27 (Elements of the Clifford group). Let (V, q) be finite dimensional quadratic vector space of dimension == dim V < oc over a fields E with char(E) * Let R V be the radical vector subspace of (V, q) with dimension r == dim R. Put m := n r 2 0. Let E1...., be an orthogonal basis of (V.q) ordered in such a way that em+l,,.- cmtrr en are the basis vectors inside R. while e1,..., Cm are spanning non-degenerate orthogonal subspace to R. inside (V, 9)-
Then every element of the Clifford group E &r(V, q) is of the form:
U1 .....gg Uk 71 7m 9,
(499)
with e E FX, k No. vi V with q(v)) # 0 foror l,...,h.
#
(500)
with Cyy E F for = I,..., .....,, m, = I,...,T, T, and some A (R).
E.4 Orthogonal Representations of the Clifford Group
Lemma E.28. Let ..,.....e en be an orthogonal basis of (V, q) and w T(V. q). If we put for j E [n]:
by == p(w)(e)))
(501)
for A C [n]:
ba = II h = (ew)
(502)
iEA
then bi...., bn is orthogonal basis for (V. q) and both (eA) se(m and (ba) are orthogonal bases for (CI(V, 4), q).
60

---

Proof. First note that .,......b, bn is a basis of V. Indeed, the releation:
n
aj bis
(503)
with d, implies:
0 (()))(()))))))))))))))))
(504)
= p(w)
Σ
04 by
(505)
p(w))-((b))
(506)
i=1
04 Bi-
(507)
Since ei...., is linear independent we get a; = 0 for all E [n]. So also by..... bn is linear independent and thus constitute basis of V.
To show that b,...,b bn an orthogonal basis of (V, q) let i + j and then consider the following:
2 b(b,,b;) byb; + byb
(508)
((w)(()))(w)(e)) + ((w)(()))((()
(509)
p(w) (exe + ejei) p(w)(2 b(ei))))
(510)
(511)
(512)
This shows that bi...., b, is orthogonal basis of (V,q). Theorem D.26 then shows that both (eл)Λε(u and (bx)Ae(m are orthogonal bases for (CI(V, q).
Theorem E.29. Let I(V.q) and CI(V,q) then we get:
((p(w)(=)) q(x).
(513)
In other words, p(w) O(CC(VV))) q). Furthermore, we have:
p(w)lA(R) id)(R).
(514)
In other words, p(w) (Crn)(((V
Proof. Let E Cn be an orthogonal basis for (V, q) and by := p(w)(e,) for j [n]. Then by Lemma E.28 know that bi...., by is an orthogonal basis of (V. q) and both (ex) and (ba)Ag(n) are orthogonal basis for (CI(VVq), q). Now let E CC(VV and write it as:
Σ AC
FA CA.
(())(===
Σ xA p(w)(ex)= AC
Σ xA bA- AçIm)
(515)
61

---

Then we get:
{((())(x)) = Σ
AA
II q(b:)
(516)
AC
iEA
Σ
AA
II [((((())):))
(517)
AC
iEA
Σ
FA II ((w)(e)))((\e))
(518)
iEA
E EA
II
p(w)(e))
(519)
AC
İEA
Σ
II p(w)(a(e) 1)
(520)
EEA
Σ
IA
II a(e)) (1))
(521)
AC
iEA
Σ IA
II ((e))
(522)
ACM
iEA
= q()).
(523)
This shows the claim. The remaining point follows from Lemma E.23
Similarly, and more detailed, we also get the following, using Theorem D.27 Lemma E.23
Corollary E.17 and
Corollary E.30. For every w I(V. q) and m .O...., n we have:
((t)) E Am
(CI" (C) (V.4) (V.g).
(524)
In words, p(w), when restricted to the m-th homogeneous multivector component C1(m)) (V, q) of CI(V.a) q) acts as an orthogonal automorphism of C1(m)) (V.9) w.r.t. q. Furthermore, it acts as the identity when further restricted the m-th homogeneous multivector component of the radical subalgebra: A (m) (R) S _C((m) (V,
Remark E.31. For in E [ni] we use (fm)) to denote the group homomorphism when restricted to on the subvector space (1(m)) (V
p (m)
r(V,9) O (R) (CI(m) VV)).
(m)) == plw)
(525)
By Theorem D.50 or Corollary E.22 we see that A [x] (R) always lies inside the kernel of pim);
A (R) C ker (m) C ((V,g).
(526)
So we get well-defined group homorphism on the quotient:
(mm)) I(V,4)/ A [x]
(R) + (m) (CI (V.4).
([w) = p(w)lermoyva (527)
Furthermore, by Theorem E.25 we have the isomorphism:
T0): :VV.9)/V///)) A [x]
(R) =
(
((w)) = p(w)lv. (528)
Consequently, for all E [n] we get the composition of group homorphisms:
f(m) OR(V,q) a r(V.4)/ A [x] (R)
(lm)
A (m))))) (R) (V, 4),
(529)
((m) = plw)) for any w E T(VV q) with (w) = Φ.
(530)
62

---

Similarly, we ger an (injective) group homomorphism:
p: (R(V q) + O (CI.)
(531)
( == p(w), for any w ((V.g) q) with p¹ (w) = =
(532)
So, the group OR (R(V acts on CI(V, q) and all subvector spaces (1(m)) (V, m = .,..., n, the same way as I ((V q) does via p. when using the surjective map pl)) to lift elements E Ox to elements w E (VV.g) q) with pl)) (w) 4.
Specifically, let x E CI(V q) be of the form x = -iir Gi + 01,1 Bakk with vas EV, a E RP and & E OR(VV.) 9) is given by 1 - ( with ET(V,q) q). then we have:
p(w)(x)) = -E c ((w)())))))) ... p(w)((L.).) SEI
Σ c ((VL)) ... iEI
(oC,))).((VV..A (VV,,,)
(533)
This means that the action e of OR (V. q) transforms a multivector z by transforming all its vector components Vi by 4 acting through the usual orthogonal transformation on vectors.
As such, we have the following equivariance property with respect to OR (v,V..).
Corollary E.32. Let F(T1..... Te+s) E F. Te+e) coefficients in F and let & E {O..... ,n.. Further, consider I elements Ti...., CI(V, q) and elements y1.....34 "y E ARR).. Then for every & E (V, q) we get the equivariance property:
be a polynomial in + 8 variables with
p(D) (F(ZI....., y1 ..........)))......... (..))......)))))) y.)(k))
(()())(1)))).....))))))(()) .... ((D)(()))))))) ... Vo)
(534)
where the superscript (k) indicates the projection onto the multivector grade-k saade---parttt of the whole expression.
E,5 The Spinor Norm and the Clifford Norm
In this subsection we shortly introduce the three slightly different versions of norm that appear in the literature: the Spinor norm, the Clifford norm and the extended quadratic form. We are interested under which conditions do they have multiplicative behaviour.
Definition E.33 (The Spinor norm and the Clifford norm). We define the Spinor norm and the Clifford norm of CI(V, q) as the maps:
SN: CI(V, 9) CI(V, 4),
SN(x) B(x)x,, CN(x) = (x)x..
(535)
CN: CI(V, - CI(V, 4)
(536)
Also recall the extended quadratic form: CI(V,q) + F.
q(x) = ((B(x)=) = C(SN(x)).
(537)
As first preliminary Lemma we need to study when the projection onto the zero-component is multiplicative:
Lemma E.34. Let E A(R) and y E CI(V, q) then we have:
<(xy) = ((x) ((v).
(538)
As result, the projection onto the zero component induces an F-algebra homomorphism:
Ç: AR) FF
y + ((y)-
(539)
Proof. We now use the notations from D.40 and Lemma D.41 A (R) and:
We distinguish two cases: x
E
A(R /^ 1) (R) A (R) = Fx
(R.
(540)
In the first case, we have: ((xy) = 0 = ((x) (yy), as multiplying with x E A (2) can only increase the grade of occurring terms or make them vanish.
In the second case, we can write x = a + f with a Fx and f f A (21) (R). Clearly, ((x) a. We then get by linearity and the first case:
{(ry) = ((ay + fv) a (()) + ((fv) (xx) ((y) + 0.
(541)
This shows the claim.
D
63

---

Lemma E.35. Let 22 E CI(V.q).
1. IfSN(a)) e 3(CI(V, q)) then we have:
= (N(n))
(542)
2. IfC((11) e 3(CI(V. q)) then we have:
=
(543)
3. IfSN(z)) E A [0] (R) or q 0 then have:
q(1122) (z)22)
(544)
Proof. SN(xi) E 3(CI(VVg)) 4)) implies:
SN(1122) (3 J21122
(545)
= ((x2) ((1))110)
(546)
((x2) (N(x1) J0z
(547)
(E(C)))) (((r))3(z2)22 SN(x:)SN(z2))
(548)
(549)
Similarly for CN.
Together with Lemma E.34 and (a1) E ^ (R) & A(R) n 3(CI(V) we get:
q(τ1σ2) = =((N((1122
(550)
-((SN()))
(551)
= =((SN(Z))) ((SN(I,)) ((SN(()))((SN(T))))
(552)
-(2))((22).
(553)
This shows the claim.
Lemma E.36. Consider the following subset of CI(V, q):
(V, == {x c1 (V, u Cull aV. Vo V 3r' V.a(z)u = d'z).
(554)
Then [V. q) is closed under multiplication and for every (V, we have: [0]
CN(2) e A (0)
SN(x) E (R).
(R).
(555)
Proof. For x, E (V, q) we also have that zy is homogeneous. Furthermore, we get for E V:
a(ry)v = a(z)a(v))
(556)
= =(z)o'y
(557)
Ezy,
(558)
for some e', E V. So, xy E (V, q) and r [ q) is closed under multiplication.
With the above conditions on we get for every u E V:
a(SN(x)) a(3()))a(z))
(559)
a(3(x))o'r
(560)
(561)
-=(((())8(0))
(562)
-a(((('2))
(563)
(((((()))))
(564)
-(a(((())))
(565)
-B(za(o))
(566)
B(xv)x
(567)
3(v)3(a)r
(568)
SN((x).
(569)
64

---

This implies by Theorem D.50 that:
SN(x) E A(R).
(570)
Since for homogeneous CI(V, q) C1 (V, q). This implies:
have: pr((3(x)) pr((#) and thus: SN(x) ((x)xx
SN (R) (V,9) A [0)
(R).
(571)
This shows the claim.
Theorem E.37 (Multiplicatiiity of the three different norms), Both, the Spinor norm and the Clifford norm, when restricted to the Clifford group, well-defined group homomorphisms:
SN: I(V.9) - A [x]
(R),
w + SN(w) = B(w)
(572)
CN: r(V,9) - A [x]
(R),
W + CN(w) = -(w)..
(573)
Furthermore, the extended quadratic form q CV(,9) q) restricted to the Clifford group is well- defined group homomorphism:
q: ((V, - R
- q(w) ((3(w)o))
(574)
Proof. This directly follows from Lemma E.35 and Lemma E.36 Note that ((V.g) C Crl-l(V. q).
Example E.38.
1. For E IF we get:
SN(a) 3(a)a a²
(575)
This also shows: i(a) = d²q(1) = a²
2. For w E V we have:
SN(w) = B(w) = = =qw).
(576)
This also shows: q(w)= = q(w).
3. For 7 1 1 eʃ with V and f E R we have:
SN(7) (1 + fe) + ef)
(577)
1 + fe eʃ + feel
(578)
1 2b(c, - q(f) q())
(579)
1.
(580)
This also shows: ((7)=
4. For g 1 + h E A (R) get:
SN(g) = -1+13(1)) 3(h)) +))
(581)
=1+ =(h)3() + h ((hh),,
(582)
and thus: q(g)= 1.
65

---

E.6 The Pin Group and the Spin Group
We have investigated the Clifford group and its action on the algebra through the twisted conjugation. In many fields of study, the Clifford group is further restricted to the Pin or Spin group. There can arise a few issues regarding the exact definition of these groups, especially when also considering fields other than the reals. We elaborate on these concerns here and leave the general definition for future discussion.
Motivation E.39 (The problem of generalizing the definition of the Spin group). For a positive definite quadratic form q on the real vector space V R" with n 2 the Spin group (pin(n) is defined via the kernel of the Spinor norm (=extended quadratic form on CI(V, q)) restricted to the special Clifford group 1 |100 (V, 9):
Spin(n) := ker q r 19) (V.q) - Bxx {W r (V.) i(w) = =1]
(v.) (1). (583)
Spin(n) is thus a normal subgroup of the special Clifford group (V, q). and, as it turns out, double cover of the special orthogonal group SO()) via the twisted conjugation p. The latter can be summarized by the short exact sequence:
(584) We intend to generalize this in several directions: from Spin to Pin group, 2. from K" to vector spaces V over general fields F with char(F) + 2, 3. from deneeeeeee to degenerate quadratic forms 9, 4. from positive (semi- Jeeinite to non-definite quadratic forms q. This comes with several challenges and umbiguities.
1 {+1} incl Spin(n) _ SO(n) -1
If we want to generalize the above to define the Pin group we would allow for elements not just of (pure) even parity w E r (V.))..). Here the question arises if one should generalize to the unconstrained Clifford group T(V q) or the (homogeneous) Clifford group T(V; q). As discussed before, to ensure that the (adjusted) twisted conjugation p is well-defined algebra automorphism of CI(V, q) the parity homogeneity assumptions is crucial. Furthermore, the elements of TV. q) that lead to non-trivial orthogonal automorphisms, e.g. v E V with q(v) + and ? =1+ ef with e E V. E RR are already homogeneous. So it is arguably safe and reasonable to restrict to the Clifford group D(V, q) and define the Pin group Pin(V, q) as some subquotient of (V..)).
In the non-definite (but still non-degenerate, real) case RR(P-),, P,q 2 1, the special orthogonal group SO(p, q) contains combinations of reflections To o T1 where the corresponding normal vectors vo. U1 E V satisfy q(vo) and q(v1) 11. Their product ug would lie in the special Clifford group plol (V,q) q). However, their spinor norm would be different from
q(vgr)) = q(vo)q(v1) = -11
(585)
Here now the question arises if we would like to preserve the former definition of Spin p. 9) as -1 ( (1) and exclude ta from Spin(p. 9), or, if we adjust the definition of Spin(p, q) and include un The former definition has the effect that Spin(p, q) does, in general, not mup surjectively onto SO(p. 9) anymore, and we would only get short exact sequence:
{+1} incl Spin(p. q) P4 *.SOpp.9) 3 R )R
(586)
(+1)
The alternative would be to define Spin q) as q (v.)) (+I). This would allow for ug Spin(p. q) and lead to the short exact sequence:
Incl
Spin(p. 4) - sO(p.g) +11,
(587)
(11)
which exactly recovers the former behaviour for Spin(n), and, which makes Spin(p, g) a double cover of SO(p. q).
However, for other fields F, cFar(F)) + 2, and non- deg-eeeeee (V. q). dim V 2 3, one would get, with the last definition -1 |vv (+1), the exact sequence:
1 1 p4 (F) incl Spin(V.q) - SO(V.q) á FK F> /FF22
(588)
66

---

and one would need to live with the fact that: a) P1(F) := E number of elements kt different from 2, rendering a (k: 1))mp in contrast to a (2 1 )map, and, again, b) the non-surjectivity of p. The latter comes from the fact that a combination of reflections To o where the corresponding normal vectors vo, U1 E V with ((v)), q(v)) + 0 cannot, in general, be normalized such that the q(ve)' 's lie in {+1} by multppyyin/ddividing the vf 'ss with some scalars E Fx. Note that we get:
= [==
could contain
q(v:/cs)=
(589)
This shows that we can only normalize the 04 's such that the q(v))'s lie inside a fixed system of representatives S C Fx Fx (F which is thus of size:
= (F* )3)
(590)
which can be different from 2 = {+
So, in this general setting, the first definition of Spin(V 9) a (1)
would at least correct the map p to be (2: 1)-mp. We would get the following short exact seequence:
[=1} incl Spin(V.q) _ SO(V,q) 3 F /1F
(591)
The normalization argument around Equation 589 for general fields E now would also give us a third option: we could, instead of restricting elements to have fixed value q(v;) E S, which depends on the choice of S, identify elements w1, w2 E r10 (V, q) if they differ by a scalar E FX:
W1 c- W2.
(592)
Then for their spinor norms (modulo (F*)²) we would get: ā(w1) = e² q(wz),
([(1))) E F=
(593)
So, the spinor norms of wn and u2 would be represented by the same representative 8 E S, as desired. However, this definition would just identify Spin(V, 9) with SO(V, 9) via p:
Spin(V, q) - qV,4))F= & SO(V.q),
(594)
and nothing new would emerge from this. Note that the latter isomorphisms always holds for non-degenerate (V.q) with dim V 3, char + 2, and can be expressed as the exact sequence:
F
incl r (0) (V.q) E SO(V.q) +
(595)
A fourth option would be to mod out the sealar squares (E*)² instead of and use (V, (V.4))( as the definition of JSSpin (Spin(V.9) 9)- This would lead the exact sequence:
(F*x)
- E (1Fx)2 incl Spin(V.q) - SO(V.9) +1
(596)
which would again coincide with the real case of Spin(n) then Rx (R (11). However, in the general case, again, p is here a (k 1)-map instead of a (2: 1)-map with k: := # (Fx )2) Furthermore, the spinor norm on Spin(V.q. q) would not be well-defined anymore, in its current form, as different representatives of elements [w1] = [wz]] E r [V) [V.q) 4)/(E*) would differ by scalar square: wi = c² W2. Their spinor norms would thus differ by by a forth scalar power:
q(w1)= q(c² wx2) ct q(w2).
(597)
So, the spinor norm on 100 (V, (V.g)/(Fx) would only be well defined modulo (FX)* C (IFX)?
(0) (V. (1 - 1F )4
[w] + [q(w).
(598)
Things become even more complicated in the degenerate case. At least we always have a short exact sequence for m 2 3:
[X] (R)
incl T(O (V,q) e SOx (V,4)
(599)
67

---

where SOx(V.q) q) indicates the set of those special orthogonal automorphisms 1 of (V.g) with 1| = idR, where R is the radical subspace of V,, 9) with r dim R. n == dim V. m == r.
Recall that we have:
[x] (R) IF*
(c] (R);
(600)
A ( 1+ span (fi ... la 2 even, fi E R, 1...., k}.
(601)
Note that for g A (R) we have p(g)lv idy and q(g) = 1. So the elements from A |-1 (R) are only blowing up the kernels of and q and can be considered redundant for our analysis. So one can argue that one can mod out A (el (R) in the above groups. We thus get short exact sequence:
F
I IV (V.9)) A (R) - SOR((..)) +11
(602)
(V..)
which now looks similar to the non-degenerate case. We can now consider the same 4 options the definition of the Spin group as before:
q (v.) (1),
q FF (V.a) (+1),
r (V.q)/F*,
[ (V.q)/(F*)).
(603)
As before, the third option can easily be discarded. If we want to preserve generality, we have the option to either preserve a and the (2 1)-property of p and pick the first option, or, preserve the surjectivity of p and take the fourth option. we are only interested in the R-case, then the second option preserves all properties. Note that in the R-case the groups of the second and forth option are isomorphic as groups anyways:
Flo (V.4) (=1) = 100 (V. (V.q)/(Rx)²
(604)
The reason is that we always have that: q(Fx) = (1*)) and, in the Rccase, the left group already contains the relevant elements to also map surjectively onto SOR (V, q) via p.
One could further discuss if one wanted replace the extended quadratic form a, which is given by q(x) = <(3(x)z) E Fx, by the other possible definition of the spinor norm SN, which is only given by SN(x) B(x)x A [x] (R). However, for all relevant elements of r (V, q) both definitions agree, but the description of the set of the rather irrelevant elements, which satisfy SN(x) = 1 and p(z)lv = idy, becomes more complicated than the set A [x] (R)) As we mod those irrelevant elements out anyways and we prefer to have our "norm map" to map to the scalars Fx instead of to A Axl (R), it is safe and reasonable to work with the extended quadratic form in all definitions.
In this paper we are mostly interested in working with the orthogonal groups and thus are interested in preserving the surjectivity of p. Since, for computational reasons, we usually restrict ourselves to the R-case, it is easier to work with (a restricted set of) elements of a group than equivalence classes, and, the spinor norn/extended quadratic form has computational meaning, we side with the second definition in this paper, but only state it for the R--case below. We leave the general definition for future discussion.
Definition E.40 (The real Pin group and the real Spin group). Let V be a finite dimensional R-vector space V, dim V = < oc, and q (possibly degenerate) quadratic form on V. We define the (real) Pin group and (real) Spin group, resp., of of q) as the following subquotients of the Clifford group T(V, q) and its even parity part T (V, q). resp.:
(+) (R).
Pin V, q) = {ur E T(V, a) ((x)) E (+1)) A
(605)
Spin (. = { f r (V. q(z) e (=1)// / (+] (R).
(606)
If (V. q) = (((..))) is the standard quadratic IR-vector space with signature q...)) then we denote: Pin(p, q, r) = Pin(R((aa.))
(607)
Spin(p, q, r) : Spin Spin (1(7))))
(608)
68

---

E.38 allow us to describe the elements of the Pin and Spin group as Corollary E.41. Let (V, q) be a finite dimensional quadratic vector space over R. Then the twisted conjugation induces weel dell-de and surjective group homomorphism onto the group of radical preserving orthogonal automorphisms of (V,9)
p: Pin(V.9) + (V,4q)
(609)
with kernel:
ker (p: Pin(V, (V, q)) {=1}.
(610)
Correspondingly, for the Spin(V, 4) group. In short, have short exact sequences:
[+1} incl Pin(V, _ (V,)) - 1,
(611)
[+1] incl Spin(V; ب SOR (V,q)
(612)
The examples E.19 E.20
follows.
Corollary E.42 (The elements of the real Pin group and the real Spin group). Let (V, q) be finite dimensional quadratic vector space over with signature (p, q, r). We get the following description of the elements of the Pin group:
101 ...1.....000 Uk 71 "pto g k No. (+1),9 A (-) (R) (R) A 1+] (R).
Pin(V.q)
(613)
where = 1 E Ej Cifepatt Cij 1,... and j I...., T, and, E V with q(v)) {+1} with =1, 1,...,l and k No. Note that {eptats l [r]] is meant span the
radical subspace R of (V,9))
We similarly can describe the Spin group follows:
Spin(V.q) toi.... 1.ii.... n "pp4 9 k E 2No q(r)) E (+1) E A (R)
/ (+] (R).
(614)
with the same conditions as above, but where needs to be an even number. Corollary E.43. Note that we also get well-defined group representations:
p: Pin(V,,) Aut ard (CI(V.)) (C(R)(C(VV
(615)
with kernel ker {=1}
In particular, CI(V, q) and (1 (V, for m O,..., are orthogonal group representations of Pin(V. q) via
p: Pin(V,q) q) + ^ tm (R) (CI (V.9).q).
(616)
Also, if F(T Te+s) E I. is a polynomial in E + variables with coefficients in RR and I.,... Te CI(V,a), yy. (RR), and [0,..., Then for every w Pin(V,q) we have the equivariance property:
p(w) (F(x.......
Iz, It,
(k) ).)) = F(p(u)(zi),,... (())( ,n, ,I,..., va) ",,,).,,,,,.
Un)
(617)
69


Reviews:
[{"review_id": "4ulsS205zo", "reviewer": "Reviewer_w3C8", "summary": "The paper develops a new kind of neural network model based on the Clifford group, which makes the neural network E(n)-equivariant.  The model is tested on a number of datasets, showing that it is capable of recovering the desired structure from the data.\n", "questions": "How would you compare to the following paper: https://proceedings.neurips.cc/paper/2021/file/f1b0775946bc0329b35b823b86eeb5f5-Paper.pdf. \nThis paper develops a universal family of neural networks for O(d)-equivariant functions. This seems to stand in tension with the claims in the Clifford algebra paper that scalar feature-based methods cannot capture certain kinds of information. (I'm sure that some more thought could explain what's going on, but I feel like that's work the authors of the submission should do, rather than the reader.) \nRelatedly, I don't think I saw any statement about whether or not the Clifford algebra approach is universal (maybe I missed it?).\nAlso, the authors end Section 3 by saying \"equivariance w.r.t. the Clifford group [...] implies equivariance w.r.t. the orthogonal group,\" but this statement, by itself, still leaves a question open: Do there exist functions equivariant under O(d) but not the \"Clifford group,\" which their method then couldn't learn?", "limitations": "n/a", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "This seems like an interesting and important direction.  The results are convincing, in the sense that this approach \"works.\"  The method is backed up by nice theory.\n\n", "weaknesses": "Although the intro and figure 1 at the beginning are helpful, I think the rest of the paper is hard to follow.  There are some theoretical results, but then when it comes to the methodology, section 4, I am left wondering what is actually being computed.  For example, what is involved in computing equation 11?  it says the x_i are elements of Cl(V,q), ok but what is the input to this system?  same for eqs. 12 and 13.  \nAnd what is the objective function?\nFor the first experiment on signed volumes, this seems like an interesting task, but it is not clear what is the form of the input data - a data cloud of x,y,z points?  What is the input to the Clifford network?\nI am having a hard time picturing concretely what is going on here.  More specifics would be helpful.\n\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the encouraging feedback. We address their concerns and questions in the following.\n\n1. *What is being computed in, e.g., equation 11?*\n\n    a. As input data, we consider points in a vector space $V$, for example, $\\mathbb{R}^3$. These can be locations, velocities, forces, etc. Since $\\mathrm{Cl}^{(1)}(V, q)$ is isomorphic to the vector space $V$ that we construct it on, we can treat the data as objects in the Clifford algebra. As such, these will be multivectors where only the vector part is nonzero. However, as we propagate through the network, the other multivector elements will get populated. As such, equation 11 computes at first linear combinations of vector coefficients expressed in a fixed basis. But later in the network, these will be multivectors. Equations 12 and 13 compute geometric products. \n\n    **Action taken**: we have added a few sentences clarifying these matters.\n\n2. *What is the objective function*? \n\n    a. The objective functions in all experiments were mean-squared errors except for the top-tagging experiment. This is a binary classification task and uses the binary cross-entropy loss. \n\n    **Action taken**: we have updated the text to clarify these matters.\n\n3. *In the volume experiments, what are the inputs to the Clifford network*?\n\n    a. Indeed, the input is a point cloud (tetrahedron). The positions are expressed as coefficients in a fixed basis, which get processed by the neural network.\n\n    **Action taken**: we have updated the text to clarify these matters.\n\n4. *Compare to Villar et al., 2021.*\n\n    a. Thanks for bringing up this paper! The crucial difference is that Villar et al.  consider representations of the orthogonal group $O(n)$  acting on $(\\mathbb{R}^{n})^k$ and maps from $(\\mathbb{R}^{n})^k \\to \\mathbb{R}^n$, or, more in our notation, $V^k \\to V$. This is a subset of the spaces that the orthogonal group can act on. There are also maps from other representations of O(n), e.g., tensor product representations, Clifford algebra representations, (irreducible) sub-representations, etc. The signed volume experiment considers a map into the space of scalars that are not invariant to reflections, which is an example of a map not captured by the theory of Villar et al.\n\n    **Action taken**: We have added this paper to the literature list and discuss our findings in comparison with these results.\n\n5. *The universality of the Clifford approach. Do there exist functions equivariant under O(d) but not the \"Clifford group\"?*\n\n    a. We show in the paper that there are maps that Clifford layers can represent that scalarization methods cannot. Similarly, we can certainly also construct maps that Clifford methods cannot represent. In our paper, we only consider one fixed action for the Clifford algebra (and their subvector spaces), namely the adjusted twisted conjugation. In this setting, $O(d)$-equivariance is the same as Clifford group equivariance. One can consider maps between spaces outside of the Clifford algebra that the Clifford group does not cover. However, the Clifford algebra offers a closed set of nice, geometrically inspired representations that are more expressive than scalars but still tractable.\n\nWe thank the reviewer again for the valuable feedback. Where applicable, we tried to be as transparent as possible about the adjustments made to the manuscript. Considering these and the above, we hope that the reviewer is open to promoting the score. If any questions or comments arise, we are happy to discuss further."}, {"review_id": "ngxowI38vH", "reviewer": "Reviewer_gQKE", "summary": "The paper introduces a new method for constructing E(n)-equivariant networks called Clifford Group Equivariant Neural Networks (CGENNs), based on an adjusted definition of the Clifford group, a subgroup of the Clifford algebra. The researchers have shown that the group’s action forms an orthogonal automorphism that extends beyond the typical vector space to the entire Clifford algebra, respecting its multivector grading and multiplicative structure. This leads to non-equivalent subrepresentations corresponding to the multivector decomposition and allows parameterizing equivariant neural network layers. CGENNs operate directly on a vector basis and can be generalized to any dimension.\n\nIncorporating group equivariance in neural networks has proved useful in many areas, including modeling the dynamics of complex physical systems and studying or generating molecules, proteins, and crystals. This paper demonstrates that the CGENNs can handle directional information more accurately and nuancedly than scalarization methods, operating directly in a vector basis rather than alternative basis representations. They can also transform higher-order features carrying vector-valued information, and readily generalize to orthogonal groups of any dimension or metric signature. The authors successfully demonstrated these advantages on various tasks, including three-dimensional n-body experiment, a four-dimensional Lorentz-equivariant high-energy physics experiment, and a five-dimensional convex hull experiment.", "questions": "1. One part that is not clear to me is how equivariance is preserved in the geometric layers if you use a linear map to first project down to $l$. This would appear to break equivariance. Can the authors please comment on this and how it doesn't break the desired symmetry?\n\n2. A limitation of the steerable equivariant literature is that it is heavily reliant on the Wigner-D matrices for $O(3)$. It seems the current effort in the Clifford group also applies to the orthogonal and $E(n)$ group. Can this approach be used to model other continuous symmetries e.g. $SU(2)$ or is this not possible?", "limitations": "N/A", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "**Originality**\nThis paper has many strengths across multiple dimensions. First, it tackles the problem of building an Equivariant network using a fresh perspective that has not been, as far as I know, used in the literature. Thus the use of the Clifford Group is an exciting direction of technological endeavor. Certainly, the mathematics behind building the Linear, Geometric Product, and Nonlinearities is nontrivial but offers what appears to be greater modeling flexibility.\n\n**Quality**\nIt is abundantly clear that immense care has been taken with the writing of this paper. This is substantiated by the large and extensive appendix which in itself can prove to be a useful reference outside of the main thesis of this paper. Furthermore, it is clear that a computational approach was taken to build the equivariant layers as I appreciated the authors that the authors acknowledged the naive $(n+1)^3$ interactions needed in their geometric layers which could be reduced by first applying a linear projection.\n\n**Clarity**\nThe authors should be given credit for attempting to make this paper clear and accessible. Unfortunately, it is not outside for those who are likely already familiar with geometric algebras. Although the writing and the flow of information are still very clear if one pretends to know what the Clifford group is. \n\n**Significance**\nIn this reviewer's opinion, this work is significant for several reasons. First, it allows for the creation of equivariant networks in a completely different manner and one that appears to be more flexible/expressive. This is due to the Clifford group allowing for non-equivalent subrepresentations. For example,  Clifford group equivariant models can extract covariant quantities which cannot be extracted from scalarization methods. Furthermore, the early empirical results suggest that such an approach also leads to empirical benefits and beats current equivariant models which suggests that this is a ripe direction for continued investigation.", "weaknesses": "This paper has a few areas for improvement that I highlight below.\n\n1.) I thank the author for their efforts to make the paper accessible, but the appendix is not a very friendly introduction. For instance, in some places, the authors acknowledge that their definitions depart from existing ones in the literature. But in other places, they claim to follow the format of other works (e.g. C.1). This makes it hard to follow and trust what the authors state.\n\n2. Perhaps what bothers me most is the fact that the motivation for using the Clifford algebra group is lost in all the technical jargon. The importance of being able to use multi-vectors in ML-specific problems is never really motivated in a convincing way. Even at present, I find it difficult to pinpoint exact non-toy applications where these tools might prove useful. Albeit there is no doubt the tools themselves are mathematically interesting.\n\n3. There are parts of the paper where it becomes unclear which part of the theory is an original contribution of the authors versus something that is already present in the math literature on Clifford group/algebra. As far as I can tell the design of the adjusted twisted conjugation is novel and its extension is an original contribution of the authors (and the subsequent ML-specific layers). Can the authors tell me if I missed anything?\n\n4. The experiments are quite preliminary. Perhaps this is by design as the authors note that the computational complexity can partially be resolved with better hardware-specific implementations. I would have loved to see the power of these methods on larger-scale equivariant problems. For instance, I am certain the authors are familiar with molecular dynamics simulations at a larger scale. These would already have the desired symmetries and established baselines to compare against.\n\n5. The authors could improve the method section by providing an empirical investigation of the runtime complexity of their actual experiments in comparison to the baselines. Right now, a reader may intuit that the Clifford group equivariant networks are more computationally expensive but, exactly how much is not immediately clear. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the actionable feedback. We address their concerns and questions in the following.\n\n1. *Accessibility and introductions of notations.*\n\n    a. During our literature research, we quickly found out that there are at least three communities (physicists, mathematicians, computer vision engineers) working with Clifford algebra (aka geometric algebra). These are all interested in different properties and, as such, use different conventions and notations that were not compatible with each-other. For example, different group actions (e.g., equation 348), or restricting themselves to homogeneous multivectors, or restricting the quadratic form to be definite or nondegenerate. One of the contributions of this paper is the study of existing literatures, unifying and generalizing them. We study the action of the orthogonal group on the whole Clifford algebra within the context of equivariant neural networks. We derived and re-derived everything we needed, with generality and group representations for NNs in mind.  \n\n2. *Motivation of operating on multivector representations.*\n\n    a. There are multiple reasons for using multivector representations. First, our theory allows for networks that operate in any inner product space, computing geometrically meaningful interactions (through geometric products). Second, if one takes a different route and goes for more general tensor product representations, one needs either irreps decomposition, which is hard to attain for any quadratic space, or use tensor product representations directly, which will grow intractably. We argue that Clifford algebra layers balances the two approaches, by allowing for a product structure (geometric product vs tensor product), which, due to their fundamental relations, will always remain finite dimensional, and, a way to decompose the whole Clifford algebra in smaller (but not necessarily irreducible) subrepresentations, which allow to reduce computational complexity (similar to the use of smaller irreps), without violating the equivariance properties. Third, there exist several physics problems that contain objects that transform in a non-standard way. For example, Maxwell’s equation treat magnetic fields as pseudovectors which can be represented as bivectors when using Clifford algebras / Clifford equivariant networks.\n    \n    **Action taken:** We elaborate on motivation for multivector representation now in more detail.\n\n3. *Originality of theory.*\n\n    a. While there already exist literature about the Clifford algebra, different notions of the Clifford groups with different actions, non of these sources we could simply cite for our purposes, because they all use their own notations/conventions, and have different corner cases, aspects or applications in mind. So a major and original contribution was to carefully construct/adjust definitions to get all our desired results (like multiplicativity, orthogonality) out and even provide proofs for the most general (non-definite, degenerate) cases.\n\n    **Action taken**: we clarify this now in the main paper.\n\n4. *Preliminary experiments.*\n    \n    a. The reason we chose not to go with molecular dynamics simulations is that the current state-of-the-art methods are highly finetuned and engineered to squeeze out all the features from the data. We felt that the adaptation and inclusion of these techniques to the Clifford layers was out of scope of the current work, and probably deserves a paper on its own. Finally, we would argue that the top-tagging experiment is at a reasonable scale and much less of a toy setting with 1.2M training samples and fully connected graphs with over 200 particles. \n    Note that architectures, such as LorentzNet, were specifically designed for this task in previous works. \n\n5. *Improve the method section by providing runtime complexity.*\n\n    a. We are hesitant with making claims about empirical runtime complexity at the current moment. The geometric product as a bilinear operation can be implemented as efficiently as existing methods that use bilinear (tensor) operations. Even stronger, this implementation would then work for any inner product space. However, comparing the current implementations to existing, highly optimized code-bases would be unfair and cast an undeserved shadow on equivariant Clifford layers. To give an approximate result. Please also consider our response to reviewer jDZi regarding runtime complexity.\n\n6. *Equivariance of projection.*\n\n    a. The equivariance of the projection follows directly from the equivariance of the linear layer. I.e., \n    $$\n    \\rho(w)\\left(T^{lin} _ {\\phi_{c_{out}}}(x _ 1, \\dots, x_\\ell)^{(k)}\\right) = \\rho(w)\\left(\\sum _ {c _ {in}=1}^\\ell \\phi _ {c _ {out}c _ {in} k} \\, x _ {c _ {in}}^{(k)},\\right)=\\sum _ {c _ {in}=1}^\\ell \\phi _ {c _ {out}c _ {in} k} \\, \\rho(w)\\left(x _ {c _ {in}}\\right)^{(k)}\n    $$\n     \n    which follows from the fact that $\\rho(w)$ is linear, commutes with scalar multiplication, and respects grade projections.\n\n7. *Use the Clifford group for other continuous symmetries like $SU(2)$.*\n\n    a. Yes, and this is very much open for exploration! For your $SU(2)$ example, see e.g. Wilson (2020). Specifics of (Clifford) group representations are elusive for now, but we believe that the presented work gives a good starting point for such scientific endeavors. As such, further explorations for Clifford networks operating in (relativistic) quantum mechanics or quantum information theory is a very interesting future direction.\n    \n\nWe are grateful for the reviewer's feedback. We have made an effort to disclose all the changes made to the manuscript as clearly as possible. Given this and our previous discussions, we are hopeful that the reviewer continues to support the paper. Should there be any questions or concerns, we are always ready for further dialogue.\n\n#### References\n- Wilson, Robert A. Subgroups of Clifford algebras. arXiv preprint arXiv:2011.05171 (2020)."}, {"review_id": "cgpqCEIN4V", "reviewer": "Reviewer_jDZi", "summary": "In this submission the authors introduce an equivariant neural network architecture building on the Clifford algebra (defined by a vector space and a quadratic form).\nThey remind that this algebra admits an orthogonal decomposition over sub-vector spaces called grade $m$ where $m$ is the number of basis elements.\nThis is referred as the 'multivector grading of the Clifford algebra'.\nThey identify a (Clifford) subgroup and its (adjusted twisted conjugation) action which not only acts on the original vector space but also on the Clifford algebra whilst satisfy many properties including multiplicatively w.r.t. the geometric product.\nThey show that equivariance w.r.t. this a Clifford group acting on the Clifford algebra implies equivariance w.r.t. the orthogonal group acting on the Clifford algebra.\nThen they propose several layers, including a linear layer and a geometric product layer which they show to be equivariant w.r.t. the action of the Clifford group.\nThen they empirically assess their proposed approach on synthetic tasks with $O(3)$, $O(5)$ or $E(3)$ equivariance and show that it overperform scalar based methods but also steerable NNs.\nFinally, they tackle the task of identifying top quarks from high-energy jets produced in particle collisions at CERN, and show that they are able to perform as well as recent specialised neural networks.\n", "questions": "- line 62: 'our method readily generalizes to orthogonal groups regardless of the dimension'. Doesn't the method proposed in [Cesa et al. 2021] also work for any dimension $n$?\n- Eq 1: What is the intuition behind the independence of the choice of orthogonal basis? Is it because by combining elements from one basis (via the geometric product) one could obtain the second basis? How come the limiting factor is really the number $m$ of such basis elements?\n- line 232: Worth saying a bit more about the 'layer-wide normalization' proposed in [70]?\n- line 235: 'However, we still require activation functions after linear layers'. Does this statement come from practical requirements for the method to work well, or from some universal approximation theorem with necessary non-linearity after the linear layer?\n- Section 5.1: Is the 'signed volume' a pseudo-scalar?\n- One of the advantage of this method that it readily works whatever the input space dimension, yet what is the scalability of the method with respect to the dimension of the space?\n- Section 5.2: What are the reasons for EMLP underperforming? It is theoretically a universal approximator.\n- Section 5: Do all methods have the same number of parameters? or same compute time?\n\n\nA PROGRAM TO BUILD E(n)-EQUIVARIANT STEERABLE CNNS, Cesa, Gabriele and Lang, Leon and Weiler, Maurice, 2021", "limitations": "- 'CGENNs induce a (non-prohibitive) degree of computational overhead'. Where does this overhead come from? What specific operation would benefit from having a custom GPU kernel? More generally it would be really nice to discuss the scalability of the method. In particular vs other 'steerable' methods which rely on parametrised tensor product via Clebsch-Gordan coefficients.\n", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "- This is a nice paper that builds on this recent research avenue of equivariant neural networks based on the Clifford/geometric algebra.\n- In find particularly appealing the fact that the methodology applies to any dimension, and more generally than $O(n)$ equivariance via the choice of quadratic form $\\mathfrak{q}$ as per Section 5.4 (if I understood correctly).\n- Additionally, it appears that the method is empirically able to fit well a variety of functions and able to generalise.\n- The paper is pretty well written given the space constraint, although I give some suggestion on improvements below.", "weaknesses": "- Likely due to space constraint, yet the methodology would deserve more discussion.\n    - For instance, how is the data embedded into the algebra (e.g. for higher order quantities such as an inertia tensor)?\n    - How is the quadratic form $\\mathfrak{q}$ chosen (e.g. is it the Minkowski product for the Jet tagging experiment)?\n    - I also believe that the linear and geometric layers could benefit from an illustrative diagram. Perhaps it may be useful to draw parallel with tensor product and irreducible representations for readers familiar with this language.\n- What's more the main paper is currently lacking a discussion on scalability of the method, and how this compares to Clebsch-Gordan tensor product approaches, although I understand that the space is limited.\n- Another points which I find slightly frustrating is the lack of ablation study, as generally it is hard to identify why this method overperform tensor product / irreps approaches. Is it more parameter efficient? Does it generalise better? Does it yield a better optimisation landscape? Is it computationally faster? Obviously fully answering this questions would deserve a full paper on its own, yet some elements of answer would be nice :)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the actionable feedback. We address their concerns and questions in the following.\n\n1. *Data embedding?*\n\n    a. The data is embedded by an additional multivector dim in the network tensors (response to reviewer W8CU), whose size depends on the algebra, e.g., 4 or 8 for 2D or 3D algebras. Multivectors, under the geom. product, parameterize bilinear operations similar to tensors. Not all tensors can be expressed as multivectors, but many from physics can. For the inertia tensor example, see Berrondo et al.(2012). \n\n    **Action taken:** we give more intuition now in the paper.\n\n2. *Choice of quadratic form?*\n\n    a. In 3D, the signature is $[1, 1, 1]$, i.e., $p=3, q=0, r=0$. For more specialized problems we use different signatures. E.g., for SU(2), we use $[-1, -1]$, i.e., $p=0, q=2, r=0$ (see response to w3C8). In space-time settings, we use $[1, -1, -1, -1]$, i.e., $p=1, q=3, r=0$. \n\n    **Action taken:** we explain this now in the paper.\n    \n3. *Visualization?*\n\n    a. We added visualizations of the layers to the attached PDF and to the appendix.\n\n4. *Scalability of the method?* \n    a.  Let $\\dim(V)=n$, then the complexity of a fully connected geometric product layer is roughly $O(c _ {out} \\cdot c _ {in}  \\cdot 2^{3n})$, where $c _ {out}$ and $c _ {in}$ are output and input channels.\n    \n    The scalability is worse than scalarization methods, and similar to Clebsch-Gordan based E3NN depending on the highest order of irrep that is chosen.\n\n5. *Performance?*\n\n    a. It is a rather general finding that any polynomial in multivectors is equivariant in any inner-product space. We constructed our layers with this in mind, ensuring that we capture the 0th, 1st, and 2nd-order terms of such a polynomial, leading to flexible parameterizations. In 3D, CGENNs are similar to E3NN based layers, i.e., existing results of E3NN based methods (e.g., TFN, SEGNN) should match the performance (see SEGNNs in the $n$-body experiment). However, depending implementation and optimization specifics, numbers can differ. Clear benefits of CGENNs are found in non-Euclidian (higher-dimensional) cases. \n\n6. *Compare to Cesa et al. (2021)?*\n\n    a. Currently, Cesa et al. support steerable CNNs equivariant to 2D/3D isometries. Other differences: Cesa et al. design equivariant CNNs, where the group acts on the grid (e.g., an image). We operate on the particles directly. Second, Cesa et al. find equivariant bases for linear maps in such CNNs. Our steerable basis is given directly by the input data, and as such we also operate using bilinear layers, which is crucial for expressive GNNs (Finzi et al., 2021).\n    \n7. *Intuition behind the independence of the choice of basis?*\n\n    a. Great question! In general, basis independency goes back to Einstein's covariance principle, stating that one should come to the same conclusions regardless of the chosen frame of reference. More specifically, it means that grade-decomposition is well-defined and not dependent on a change of coordinates. We were unable to find in the literature a proof for this decomposition in the (non-definite, potentially degenerate) general case. As such, we regard it an original contribution of the work. Secondly, it directly allows for the equivariance of the grade projections! \n\n    **Action taken**: we explain the basis independence proof more clearly now.\n\n8. *Layer norm.*\n\n    a. The layer norm is defined as $x \\mapsto \\frac{x - \\mathbb{E}[x]}{\\mathbb{E}[\\sqrt{|q(x)|}}$, where the expectations are taken over channel dims. Intuitively, it re-centers and rescales multivectors according to their average norm.\n\n9. *Nonlinearities?*\n    \n    a. We need a sufficiently flexible, nonlinear function to learn expressive representations. We empirically find that including additional nonlinearities after the linear layers results in improved performances.\n\n    **Action taken**: we mention this in the methodology section now.\n\n10. *Signed-volume a pseudoscalar?*\n    \n    a. Yes.\n\n11. *Scalability regarding space dimension?*\n\n    a. The dimensionality of the algebra scales as $2^n$, where $n=\\dim(V)$. This looks bad at first sight, but usually the dimensionality of geometric spaces is not so large. Further, one can operate using subalgebras, which is typical in geometric algebra literature, or one can simply not consider the highest subspaces. \n\n12. *What are the reasons for EMLP underperforming?* \n    a. For O(5) regression, we took datasets and EMLP results directly from Finzi et al. (2021). We went through the EMLP code and noted that the output irreps of each layer is set equal to the output irrep of the whole task. For the O(5) regression task, whose output irrep is one-dimensional, this means that there will be no vector-valued outputs of the bilinear layers. This is potentially less expressive. Further, it depends highly on the specifics of the implementations and parameterizations. Our polynomials form a rather general basis from which we constructed our layers. \n\n13. *Do all methods have the same number of parameters or compute-time?*\n    \n    a. We kept the number of parameters similar between models. The compute-time however can depend on which method one uses. Steerable methods that use bilinear layers are generally slower than scalarization methods. Compared to existing steerable methods, Clifford layers are not fundamentally less efficient. \n\n14. *Computational overhead, scalability.*\n\n    a. The computations overhead originates from the bilinear geometric product layers. Please also consider our previous responses (4, 13) regarding scalability.\n\n#### Refs\n- Berrondo, M., J. Greenwald, and C. Verhaaren. Unifying the inertia and Riemann curvature tensors through geometric algebra. American Journal of Physics 80.10 (2012): 905-912.\n- Finzi, Marc, Max Welling, and Andrew Gordon Wilson. A practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups. ICML, 2021."}, {"review_id": "TOgKOM4QwL", "reviewer": "Reviewer_W8CU", "summary": "I like to admit that I am not familiar with this area of work. I try my best to review the key contributions of this work. My review is more of clarification than critic.\n\nThe theory part of the paper lies mainly in section 3.1 starting with Eq.(5).\nThe computation part of the paper is given in section 4. The authors introduce two kinds of layers. Linear Layers and Geometric Product Layers. \n\nLinear layer mix the input channels of the same grade (Eq.(11)).\nInteraction layer takes the geometric product of the i and j grade of x1, x2 and project to the k-th grade.\n\nSeveral experiments were performed using this neural network. I will discuss more about the experiments in later part of this review.", "questions": "How to prepare data to feed into the neural network? Perhaps the authors can give a simple example(s) of processing complex numbers and/or quaternions.\n", "limitations": "Readers without a background of abstract algebra or Clifford algebra may find it hard to understand this paper.", "rating": 8, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper explore a different approach to deep learning. This paper helps in adding a new dimension in the field of deep learning.\n2. The authors provided a huge amount of background and foundations in the supplementary materials. This is important because I presume relatively few readers are familiar with Clifford algebra.\n3. Although I cannot fully understand the theoretical part of this work. It seems to me the authors try to substantiate their work with theories.", "weaknesses": "1. The strength of this paper may also be \"its weakness\", readers without background on Clifford algebra may struggle to pick up this area of work. It will be good if the authors follow up with a good tutorial. Perhaps a series of online learning materials and provide its link in this paper. For example, make an archive tutorial paper (or online video) and those locations are cited in this paper. The supplementary materials of this paper is good. but it is insufficient to bring people outside of this field up to the level to use this paper.\n2. The output of the neural network is given by Eq.(11) and Eq.(13). I understand that the output is also an element of the Clifford algebra. I struggle to understand how this output is link to the use cases in the experiments. It will be good if the authors help the readers by describing the experiments in detail. Give a step-by-step walk through of :\n2a: how the data is being prepared and in what format\n2b: how the data is being feed into the neural network\n2c: how to link the k-th grade of x to the numerical output.\n2d: perhaps walk through with a simple example, like one example using complex numbers.\n\nI explain 2c a bit more here. For a complex number c1 = 2+3i, the zeroth grade is 3 (or the 1 element). The first grade is 2 (or i). If the author can guide the reader through on how to input c1 into the linear layer and explain what is the output, that will help. Similarly, the author may also give another example c1=1-2i and c2=4+I on how to feed c1 and c2 into Eq.(12) and Eq.(13).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the actionable feedback. We address their concerns and questions in the following.\n\n1. *Familiarize readers with the Clifford algebra, perhaps with a tutorial.*\n\n    a. Following up with a tutorial is a great suggestion! Indeed, we are already working on a blog post series that explains the subject matter in a less demanding way. We realized, after studying related work, that the orthogonal group acting on the Clifford algebra was not studied in a most general form. Physicists, engineers, and mathematicians usually restricted themselves to, e.g., definite or nondegenerate quadratic forms. In the context of deep learning, we studied this more generally, requiring a more rigorous approach. \n\n2. *How is the data being prepared and processed? Explain with an example.*\n\n    a. We can give you some intuitions here. 2a: The neural network tensor shapes of the Clifford algebra networks are of the form $B \\times C \\times 2^n$, where $n = \\dim(V)$. If we have scalar-valued data, we can use the first component of that last dimension to embed those, since $\\mathrm{Cl}^{(0)}(V, q)=\\mathbb{F}$. For vector-valued data we can use $\\mathrm{Cl}^{(1)}(V, q) = V$. The other components are left at zero at first. 2b: This tensor is then fed into the neural network, which also computes features in the other Clifford subspaces, and as such, we get densely filled multivectors. 2c: At the end of the network, we can project onto the grade $k$ subspace to get a $k$-vector. Usually, this would be the grade $0$ subspace for scalar-valued predictions or grade $1$ for vector-valued predictions. 2d: Regarding your complex numbers example, the tensor shape would be $B \\times C \\times 2$, where we have 2 for the real and imaginary components. You populate the last dimension using your complex-valued data. We hope this helps!\n    \n    **Action taken:** we now give a bit more intuition about these procedures in the experimental section.\n\n\nWe thank the reviewer for their valuable insights. We have strived to be as explicit as possible in tackling the remaining issues. If our responses meet expectations, we hope the reviewer maintains their support for our paper. We are ready for any more discussions or questions the reviewer might have."}, {"review_id": "D7kGEppIrO", "reviewer": "Reviewer_5NZ6", "summary": "This paper presents a novel approach for creating E(n)-equivariant models based on Clifford (geometric) algebras. \n\nThe authors identify the Clifford group and its action. By extending this action to the entire Clifford algebra, remarkable properties can be obtained enabling the construction of equivariant maps from the Clifford algebra to itself, as well as other equivariant operations, such as grade projection. \nUsing these, an equivariant Clifford group neural network operating on vector fields can be constructed, capable of a more accurate and nuanced interpretation of the underlying structures than the baseline scalar-field networks. \n\nThe proposed method exhibits application versatility across different tasks and dimensionalities of the space.", "questions": "Q1. Translations: \n\n- The authors claim to present an E(n)-equivariant method.  \n\n- As discussed in Section 3.2, equivariance w.r.t. the Clifford group implies equivariance w.r.t. the orthogonal group O(n). It is, however, unclear how **translation** equivariance is attained. \nIn fact, all the presented experiments involve only the orthogonal group transformations: in the $n$-body experiment, the authors use mean-subtracted positions of the particles, thereby removing the effect of translation already on the input level. \n\n- Could the authors clarify how translations can be handled using the proposed layers? \n\n \n\nQ2. How are O(n)-**invariant** predictions obtained with the proposed method? \n\n \n\nQ3. The claim of better underlying geometric properties capturing is meant to be supported by the *signed-volume experiment*. I would like for the authors to elaborate on the claim itself and clarify the presented experimental support. \n\n- How do the authors define *covariance*? \n\n- Some of the considered scalar-feature methods extract equivariant features and compute the invariant features from them to perform classification/segmentation requiring this. \nFor instance, in the theory of the VN method [34], there's nothing that restricts the features to only be equivariant under rotations and not reflections. In fact, the invariant features are obtained as the inner product of equivariant features, which also cancels the effect of reflections thus making the method produce O(3)-invariant predictions.\nThis makes the VN method unable to distinguish a tetrahedron and its reflected copy (and thus, their signed volumes will not be distinguished either) by the design of the invariant computation block only, even though the method extracts equivariant features. \nThis should be taken into account in the comparison in Section 5.1. \n\n- Could the authors provide more details of the experimental setup? I want to see the number of samples, the ratio of positive/negative volumes, and the success rates of the predictions given positive/negative volumes for the proposed method and the baselines.\n\n \n\nQ4. Could the authors provide details of their method architecture used in the experiments in Sections 5.1-2? How does the complexity of the method compare to the baselines? \n\n \n\nQ5. Is the training setup the same for all the methods in the experiments? I.e., were the training hyperparameters optimized for the proposed method and used for others? ", "limitations": "The limitations and broader impact are adequately addressed by the authors. ", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "S1. The high quality of writing and presentation are two strong features of the paper. The authors succeeded in finding a good balance between the amount of technical detail and the narration clarity. \n\nS2. The paper presents novel theoretical results based on Clifford algebras, enabling an original way of constructing equivariant neural networks, which is a great contribution to the field.  \n\nS3. Experimental validation is versatile and, to a large extent, convincing.", "weaknesses": "W1. Missing details in the presentation of the proposed method, including: \n\n(See the questions section for more information.) \n\n- translation equivariance (Q1) \n\n- invariant prediction computation (Q2), \n\n- the architecture of the proposed networks in the experiments (Q4). \n\n \n\nW2. Incomplete comparison with other methods:   \n\n- The support of the claim that the scalar-feature methods are not able to extract some essential geometric properties from input data needs to be elaborated on (Q3). \n\n- Discussion on the complexity of the proposed method vs the baselines is missing (Q4).   \n\n- Some training details are missing (Q5). \n\n\nAddressing these weaknesses will improve the presentation and clarify the support of certain claims. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their highly valuable feedback. We address their concerns and questions in the following. The reviewer has already transformed weaknesses into questions (thanks!), which we answer directly.\n\n1. *How is translation equivariance obtained?*\n\n    a. Translation invariance is obtained through the subtraction of a reference point, like the mean position of the point cloud, which is a typical approach to getting translation invariance. To obtain equivariance, one simply adds the reference point back onto the NN's prediction. Villar et al. (2021) show that this has some universality properties regarding invariant functions. While relative positions are the golden standard for translation equivariance, the Clifford algebra approach can offer an alternative. One of the reasons why we developed the theory for the fully general, potentially degenerate metric case is that we can get $\\rho(w)$ to act as a Euclidean isometry (see Remark D.30). This means we can get translation equivariance without subtracting a reference point. This is what the recent developments in projective geometric algebra (PGA) study, which use the signature $(n, 0, 1)$ (Roelfs \\& de Keninck, (2021)). However, there were some practical technicalities. To give an idea, we have to map the data from the usual vector space V to its dual subspace in a well-defined manner, which is where the Euclidean group acts. However, here the metric is fully degenerate and does not correspond to the metric on the original space V. These difficulties motivated us to leave this for future work.\n2. *How are invariant predictions obtained?*\n\n    a. There are two approaches to obtaining invariance. First, the scalar subspace $\\mathbb{F}=\\mathrm{Cl}^{(0)}(V, q)$ is always invariant with respect to $\\rho(w)$. So, one can take the grade projection of the NNs output onto the scalar subspace to obtain invariant outputs. On top of this, the algebra-extended quadratic form $\\bar{q}: \\mathrm{Cl}(V, q) \\to \\mathrm{Cl}^{(0)}(V, q)=\\mathbb{F}$ is always invariant. So one can concatenate the zero-projection with the quadratic form of the covariant subspaces to get an invariant. One can use typical feed-forward neural networks to project this to an invariant of the dimension the problem requires.\n    \n    **Action taken:** we now clarify this in the invariant tasks.\n\n3. *Clarifications regarding the volume experiments.*\n\n    a. 1. We define a covariant object as a quantity that transforms in a nontrivial but predictable way under a change of coordinates. 2. The reviewer is right that scalar methods such as the default implementation of VNs produce O(3) invariant features, and thus VNs are not able to distinguish a tetrahedron and its reflected copy. We add this for clarity. Of course, we do not claim that such methods cannot be adapted to obtain access to these features. However, one of the merits of the Clifford algebra networks is that the network can readily distinguish these cases. 3. We're sorry that there is an unclarity regarding classification/regression. In Sec. 5.1, we detail that we regress the signed volume against the spatial positions. The number of training samples is then given in Figure 3. The ratio of positive vs. negative volumes is 50/50. The task is thus to predict the volume and not only its orientation.\n\n    **Action taken:** we adjusted section 5 to clarify these matters.\n\n4. *Clarify the neural architecture against baselines for Section 5.1-2. How does the complexity of the method compare against the baselines?*\n\n    a. We use feed-forward networks based on the layers proposed in Section 3. We compare against baselines of similar numbers of parameters, where source codes were taken from original implementations. For 5.2, we took the numbers directly from the source paper, and used networks with similar numbers of parameters. For further details, please consider the attached PDF file.\n\n    **Action taken:** we now elaborate better on model and baseline complexities in the experimental section.\n\n5. *Is the training setup the same for all experiments? Were hyperparameters optimized?*\n    a. We did no exessive hyperparameter tuning for CGENN architectures in experiments 5.1 and 5.2. We rather used design principles (number of parameters, layers from baseline architectures). We searched for learning rates for 5.3 and tried a few architectures and learning rate regimes for 5.4. Reported baseline numbers for experiments 5.2, 5.3, and 5.4 are taking from existing literature. For experiments in 5.1, baselines were optimized for similar parameter counts as CGENNs. We also ran small learning rate searches for them. \n\nWe are very grateful for the reviewer's feedback. Where applicable, we have made an effort to disclose all the changes made to the manuscript. Given this and our previous discussions, we are hopeful that the reviewer is open to promoting the score. Should there be any questions or concerns, we are always ready for further dialogue.\n\n#### References\n- Villar, Soledad, et al. Scalars are universal: Equivariant machine learning, structured like classical physics. Advances in Neural Information Processing Systems 34 (2021): 28848-28863.\n- Roelfs, Martin, and Steven De Keninck. Graded symmetry groups: plane and simple. Advances in Applied Clifford Algebras 33.3 (2023): 30."}]

Author rebuttal:
We extend our gratitude to the reviewers for their efforts in evaluating our paper. The overall positive response we received has been highly encouraging. The feedback has undoubtedly improved the quality of our work, and we sincerely appreciate the insightful comments and suggestions that were provided.

We endeavored to consider the feedback as comprehensively as possible,  leading to a revision process that significantly honed the paper. We have strived to make all the necessary changes, balancing them with given space constraints. We have addressed every point in our responses and indicated where the paper was adapted based on the feedback.

We would like to reiterate that our work on Clifford Group Equivariant Neural Networks (CGENNs) sets a distinctive path from the usual geometric deep learning approaches. Unlike typical methods, CGENNs operate directly on an input vector basis compared to, e.g., popular spherical harmonics basis, are more expressive than scalar methods, and apply to inner product spaces of any dimension. 

Concluding, we thank the reviewers for recognizing the potential and novelty of our work. It is encouraging to know that our paper is viewed as a significant contribution that provides a direction in the field of geometric deep learning. 

Meta-review: This paper considers the Clifford Algebra and designs linear and polynomial layers that are equivariant to actions from the Clifford group. This allows designing neural network architecture suitable for different geometric settings including estimation of volume, convex hull, and $E(3)$ equivariant functions. We encourage the authors to make an effort improving the readability and accessibility of the paper to the general GDL community. Making it more self-contained and adding relevant implementation details would be appreciated.
Decision: Accept (oral)