SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Theorem 4.1 (Universality of  $G$ -MACE). Assume that the one-particle embedding  $(\phi_k)_k$  is a complete basis. Then, the set of  $G$ -MACE models, with a fixed finite number of layers  $T$ , is dense in the set of continuous and equivariant properties of point clouds  $X \in \mathrm{msets}(\Omega)$ , in the topology of pointwise convergence. It is dense in the uniform topology on compact and size-bounded subsets.
5 lie-nn: Generating Irreducible Representations for Reductive Lie Groups
In order to construct the G-cluster expansion for arbitrary Lie groups, one needs to compute the generalized Clebsch-Gordan coefficients (10) for a given tuple of representations (see 11). To facilitate this task, we have implemented an open source software library,  $\mathrm{lie - nn}^1$ . In this section we review the key techniques employed in this library.
1https://github.com/xxx/xxx
5.1 Lie Algebras of Reductive Lie Groups
Formally, the Lie algebra of a Lie group is its tangent space at the origin and carries an additional structure, the Lie bracket. Informally the Lie algebra can be thought of as a linear approximation to the Lie group but, due to the group structure, this linear approximation carries (almost) full information about the group. In particular the representation theory of the Group is almost entirely determined by the Lie algebra, which is a simpler object to work with instead of the fully nonlinear Lie group.
Lie algebra The Lie groups we study can be realized as closed subgroups  $G \subset \mathrm{GL}_n(\mathbb{R})$  of the general linear group. In that case their Lie algebras can be concretely realized as  $\mathfrak{g} = \operatorname{Lie}(G) = \{X \in M_n(\mathbb{R}) \mid \forall t \in \mathbb{R} : \exp(tX) \in G\}$  where  $\exp(X) = 1 + X + \frac{1}{2} X^2 \ldots$  is the standard matrix exponential. It turns out that  $\mathfrak{g} \subset M_n(\mathbb{R})$  is a linear subspace closed under the commutator bracket  $[X, Y] = XY - YX$ .
Structure theory We fix a linear basis  $\{X_i\} \subset \mathfrak{g}$ , called a set of generators for the group. The Lie algebra structure is determined by the structure constants  $A_{ijk}$  defined by  $[X_i, X_j] = \sum_k A_{ijk} X_k$ , in that if  $X = \sum_i a_i X_i$  and  $Y = \sum_j b_j X_j$  then  $[X, Y] = \sum_k \left( \sum_{i,j} A_{ijk} a_i b_j \right) X_k$ . The classification of reductive groups provides convenient generating sets for their Lie algebras (or their complexifications). One identifies a large commutative subalgebra  $\mathfrak{h} \subset \mathfrak{g}$  (sometimes of  $\mathfrak{g}_{\mathbb{C}} = \mathfrak{g} \otimes_{\mathbb{R}} \mathbb{C}$ ) with basis  $\{H_i\}$  so that most (or all) of the other generators  $E_\alpha$  can be chosen so that  $[H_i, E_\alpha] = \alpha(H_i) E_\alpha$  for a linear function  $\alpha$  on  $\mathfrak{h}$ . These functions are the so-called roots of  $\mathfrak{g}$ . Structural information about  $\mathfrak{g}$  is commonly encoded pictorially via the Dynkin diagram of  $\mathfrak{g}$ , a finite graph the nodes of which are a certain subset of the roots. There are four infinite families of simple complex Lie algebras  $A_n = \mathfrak{su}(n+1)$ ,  $B_n = \mathfrak{so}(2n+1)$ ,  $C_n = \mathfrak{sp}(2n)$ ,  $D_n = \mathfrak{so}(2n)$  and further five exceptional simple complex Lie algebras (a general reductive Lie algebra is the direct sum of several simple ones and its centre). The Lie algebra only depends on the connected component of  $G$ , thus when the group  $G$  is disconnected in addition to the infinitesimal generators  $\{X_i\}$  one also needs to fix so-called "discrete generators", a subset  $\mathbf{H} \subset G$  containing a representative from each connected component.
Figure 2: Examples of Dynkin diagrams and their associated group class.
Representation theory The representation theory of complex reductive Lie algebras is completely understood. Every finite-dimensional representation is (isomorphic to) the direct sum of irreducible representations ("irreps"), with the latter parametrized by appropriate linear functional on  $\mathfrak{h}$  ("highest weight"). Further given a highest weight  $\lambda$  there is a construction of the associated irrep with an explicit action of the infinitesimal generators chosen above. The Weyl Dimension Formula gives the dimension of an irrep in terms of its highest weight.
5.2 Numerical Computations in lie-nn
The most basic class of the lie-  $\mathfrak{m}$  library encodes a group  $G$  and infinitesimal representation  $d\rho$  of  $\mathfrak{g}$  using the tuple
$$
\rho := (A, n, \{d \rho (X _ {i}) \} _ {i}, \{\rho (h) \} _ {h \in \mathbf {H}}), \tag {20}
$$
with  $A$  the structure constants of the group,  $n$  the dimension of the representation, and  $d\rho(X_i)$  and  $\rho(h)$  being  $n \times n$  matrices encoding the action of the infinitesimal and the discrete generators respectively. The action of infinitesimal generators is related to the action of group generators by the exponential,  $\forall X \in \mathfrak{g}, \rho(e^X) = e^{d\rho(X)}$ . For finite groups, we assume that  $d\rho(X) = 0$  as they have only discrete generators.
As the building blocks of the theory irreps are treated specially; the package implements functionality for the following operations for each supported Lie group:
- Constructing the irrep with a given highest weight.
- Determining the dimension of an irrep.
- Decomposing the tensor product of several irreps into irreps up to isomorphism (the selection rule, giving the list of irreducible components and their multiplicities).
- Decomposing the tensor product of several irreps into irreps explicitly via a change of basis ("generalized Clebsch-Gordan coefficients").
- Computing the symmetrized tensor product of the group (see. 5.3 and A.9 for details).
To construct an irrep explicitly as in (20) one needs to choose a basis in the abstract representation space (including a labeling scheme for the basis) so that we can give matrix representations for the action of generators. For this purpose, we use in lie-nn the Gelfand-Tsetlin (GT) basis (Gelfand and Tsetlin, 1950) and associated labeling of the basis by GT patterns (this formalism was initially introduced for algebras of type  $A_{n}$  but later generalized to all classical groups). Enumerating the GT patterns for a given algebra gives the dimension of a given irrep, the selection rules can be determined combinatorially, and it is also possible to give explicit algorithms to compute Clebsch-Gordan coefficients (the case of  $A_{n}$  is treated by Alex et al. (2011)). For some specific groups, simplifications to this procedure are possible and GT patterns are not required.
In some cases, one wants to compute coefficients for reducible representations or for representations where the analytical computation with GT patterns is too complex. In these cases, a numerical algorithm to compute the coefficients is required. Let  $d\rho_{1}, d\rho_{2}$  be two Lie aglebra representations of interest. The tensor product on the Lie algebra  $d\rho_{1} \otimes d\rho_{2}(X)$  can be computed as,
$$
d \rho_ {1} \otimes d \rho_ {2} (X) = d \rho_ {1} (X) \otimes 1 + 1 \otimes d \rho_ {2} (X) \tag {21}
$$
Therefore, given sets of generators of three representations  $d\rho_{1}, d\rho_{2}, d\rho_{3}$ , the Clebsch-Gordan coefficients are the change of basis between  $(d\rho_{1}(X) \otimes 1 + 1 \otimes d\rho_{2}(X))$  and  $d\rho_{3}(X)$ . One can compute this change of basis numerically via a null space algorithm. For some groups, one can apply an iterative algorithm that generates all irreps starting with a single representation, using the above-mentioned procedure (see A.10).
5.3 Symmetric Powers
Let  $V$  be a vector space and  $\{e_i\}$  be a basis of  $V$ . The symmetric power of  $V$ ,  $\operatorname{Sym}^n V$ , can be regarded as the space of homogeneous polynomials of degree  $n$  in the variables  $e_i$ . The product basis in Equation 10 spans exactly this space. A basis of  $\operatorname{Sym}^n V$  can be constructed as,
$$
\left\{e _ {i _ {1}} \cdot e _ {i _ {2}} \dots e _ {i _ {n}} \mid i _ {1} \leq \dots \leq i _ {n} \right\} \tag {22}
$$
If  $V_{\lambda}$  if an irreducible representation of a reductive Lie group  $G$  with highest weight  $\lambda$ ,  $\operatorname{Sym}^n V_{\lambda}$  admits a decomposition into irreducible representations,
$$
\operatorname {S y m} ^ {n} V _ {\lambda} = \bigoplus c _ {\lambda , \mu} V _ {\mu} \tag {23}
$$
The generalized Clebsch-Gordan coefficients in (11) represent the change of basis between  $\mathrm{Sym}^n V_\lambda$  and one of the  $V_{\mu}$ . The following steps are taken to obtain these coefficients:
- Construct the symmetric power basis as in (22)
- Compute the coefficients  $c_{\lambda, \mu}$ , using Freudenthal's Formula or GT patterns.
- For any  $\mu$  with  $c_{\lambda, \mu}$  non-zero, find a basis of  $V_{\mu}$ , and compute the change of basis between the basis of  $\operatorname{Sym}^n V_{\lambda}$  and  $V_{\mu}$
Alternatively, if one simply has the Clebsch-Gordan coefficients, the change of basis from  $V_{\lambda}$  to some  $V_{\mu}$ , a new algorithm outlined in Appendix A.9 and implemented in Lie-nn can construct the change of basis from  $\mathrm{Sym}^n V_{\lambda}$  to  $V_{\mu}$ .
6 Applications
6.1 Lie groups and their applications
In Table 6.1 we give a non-exhaustive overview of Lie groups and their typical application domains, to which our methodology naturally applies.
Benchmarking our method on all of these applications is beyond the scope of the present work, in particular, because most of these fields do not have standardized benchmarks and baselines to
Table 1: Lie groups of interests covered by the present methods and their potential applications to equivariant neural networks. The groups above the horizontal line are already available in lie-nn. The ones below the line fall within our framework and can be added.
Group	Application	Reference
U(1)	Electromagnetism	(Lagrave et al., 2021)
SU(3)	Quantum Chromodynamics	(Favoni et al., 2022)
SO(3)	3D point clouds	(Batatia et al., 2022a)
SO+(1,3)	Particle Physics	(Bogatskiy et al., 2020b)
SL(3,R)	Point cloud classification	-
SU(2N)	Entangled QP	-
Sp(N)	Hamiltonian dynamics	-
SO(2N+1)	Projective geometry	-
compare against. The MACE architecture has proven to be state of the art for a large range of atomistic modeling benchmarks (Batatia et al., 2022a). In the next section, we choose two new prototypical applications and their respective groups to further assess the performance of our general approach.
6.2 Particle physics with the  $SO(1,3)$
Jet tagging consists in identifying the process that generated a collimated spray of particles called a jet after a high-energy collision occurs at particle colliders. Each jet can be defined as a multiset of four-momenta  $[(E_i,\mathbf{p}_i)]_{i = 1}^N$ , where  $E_{i}\in \mathbb{R}^{+}$  and  $\mathbf{p}_i\in \mathbb{R}^3$
Current state-of-the-art models incorporate the natural symmetry arising from relativistic objects, e.g., the Lorentz symmetry, as model invariance. To showcase the performance and generality of the  $G$ -MACE framework we use the Top-Tagging dataset (Butter et al., 2019), where the task is to differentiate boosted top quarks from the background composed of gluons and light quark jets. In Table 6.2, we can see that  $G$ -MACE achieves excellent accuracy, being the only arbitrary equivariant model to reach similar accuracy as PELICAN, which is an invariant model. We refer to Appendix A.11.1 for the details of the architecture.
Table 2: Comparisson between state-of-the-art metrics on the Top-Tagging dataset. Scores were taken from (Bogatskiy et al., 2022; Qu et al., 2022; Qu and Gouskos, 2020; Munoz et al., 2022; Bogatskiy et al., 2020a; Komiske et al., 2019; Pearkes et al., 2017).
Architecture	#Params	Accuracy	AUC	Rej30%
PELICAN	45k	0.942	0.987	2289 ± 204
partT	2.14M	0.940	0.986	1602 ± 81
ParticleNet	498k	0.938	0.985	1298 ± 46
LorentzNet	224k	0.942	0.987	2195 ± 173
BIP	4k	0.931	0.981	853 ± 68
LGN	4.5k	0.929	0.964	435 ± 95
EFN	82k	0.927	0.979	888 ± 17
TopoDNN	59k	0.916	0.972	295 ± 5
LorentzMACE	228k	0.942	0.987	1935 ± 85
6.3 3D Shape recognition
3D shape recognition from point clouds is of central importance for computer vision. We use the ModelNet10 dataset (Wu et al., 2015) to test our proposed architecture in this setting. As rotated objects need to map to the same class, we use a MACE model with  $O(3)$  symmetry. To create an encoder version of  $G$ -MACE, we augment a PointNet++ implementation (Yan, 2019) with  $G$ -MACE layers. See the appendix A.11.2 for more details on the architecture. We see in Table 3 that MACE outperforms the non-equivariant baseline.
Table 3: Accuracy in shape recognition. Scores were taken from (Qi et al., 2016), (Qi et al., 2017), Shen et al. (2018), Li et al. (2018), Kumawat and Raman (2019)
Architecture	PointMACE (ours)	PointNet	PointNet ++	KCN	SO-Net	LP-3DCNN
Accuracy	96.1	94.2	95.0	94.4	95.5	94.4
Representation	Point Cloud	Point cloud	Point cloud	Point Cloud	Point Cloud	Voxel grid
We introduced the  $G$ -Equivariant Cluster Expansion, which generalizes the successful ACE and MACE architectures to symmetries under arbitrary reductive Lie groups. We provide an open-source Python library lie-nn that provides all the essential tools to construct such general Lie-group equivariant neural networks. We demonstrated that the general  $G$ -MACE architecture simultaneously achieves excellent accuracy in Chemistry, Particle Physics, and Computer Vision. Future development will implement additional groups and generalize to new application domains.
IB's work was supported by the ENS Paris Saclay. CO's work was supported by NSERC Discovery Grant IDGR019381 and NFRF Exploration Grant GR022937. This work was also performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3). IB would like to thank Gábor Csányi for his support.
Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel, Neural Computation 1, 541 (1989).
T. S. Cohen and M. Welling, ICLR 2017 (2016a), 10.48550/ARXIV.1612.08498.
C. Esteves, C. Allen-Blanchette, X. Zhou, and K. Daniilidis, "Polar transformer networks," (2017).
B. Anderson, T. S. Hy, and R. Kondor, in Advances in Neural Information Processing Systems, Vol. 32, edited by H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett (Curran Associates, Inc., 2019).
A. P. Bartók, R. Kondor, and G. Csányi, Physical Review B 87, 184115 (2013).
S. Batzner, A. Musaelian, L. Sun, M. Geiger, J. P. Mailoa, M. Kornbluth, N. Molinari, T. E. Smidt, and B. Kozinsky, Nature Communications 13, 2453 (2022).
I. Batatia, D. P. Kovács, G. N. C. Simm, C. Ortner, and G. Csányi, "Mace: Higher order equivariant message passing neural networks for fast and accurate force fields," (2022a).
A. Bogatskiy, T. Hoffman, D. W. Miller, and J. T. Offermann, Machine Learning and the Physical Sciences workshop, NeurIPS 2022 (2022), arXiv:2211.00454 [hep-ph].
C. Li, H. Qu, S. Qian, Q. Meng, S. Gong, J. Zhang, T.-Y. Liu, and Q. Li, (2022), arXiv:2208.07814 [hep-ph].
M. Finzi, S. Stanton, P. Izmailov, and A. G. Wilson, in Proceedings of the 37th International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 119, edited by H. D. III and A. Singh (PMLR, 2020) pp. 3165-3176.
R. Kondor and S. Trivedi, in Proceedings of the 35th International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 80, edited by J. Dy and A. Krause (PMLR, 2018) pp. 2747-2755.
A. Bogatskiy, B. Anderson, J. T. Offermann, M. Roussi, D. W. Miller, and R. Kondor, "Lorentz Group Equivariant Neural Network for Particle Physics," (2020a), arXiv:2006.04780 [hep-ph].
R.Drautz,Phys.Rev.B 99,014104 (2019).
I. Batatia, S. Batzner, D. P. Kovács, A. Musaelian, G. N. C. Simm, R. Drautz, C. Ortner, B. Kozinsky, and G. Csányi, "The design space of e(3)-equivariant atom-centered interatomic potentials," (2022b).
T. Cohen and M. Welling, in Proceedings of The 33rd International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 48, edited by M. F. Balcan and K. Q. Weinberger (PMLR, New York, New York, USA, 2016) pp. 2990-2999.
T. S. Cohen, M. Geiger, J. Kohler, and M. Welling, in International Conference on Learning Representations (2018).
M. Weiler, P. Forre, E. Verlinde, and M. Welling, "Coordinate independent convolutional networks - isometry and gauge equivariant convolutions on riemannian manifolds," (2021).
G. Dusson, M. Bachmayr, G. Csányi, R. Drautz, S. Etter, C. van der Oord, and C. Ortner, Journal of Computational Physics 454, 110946 (2022).
M. Geiger and T. Smidt, "e3nn: Euclidean neural networks," (2022).
M. Finzi, M. Welling, and A. G. Wilson, "A practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups," (2021).
R. Kondor, Z. Lin, and S. Trivedi, in Advances in Neural Information Processing Systems, Vol. 31, edited by S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (Curran Associates, Inc., 2018).
V. G. Satorras, E. Hoogeboom, and M. Welling, "E(n) equivariant graph neural networks," (2021).
J. Brandstetter, R. Hesselink, E. van der Pol, E. J. Bekkers, and M. Welling, "Geometric and physical quantities improve e(3) equivariant message passing," (2022), arXiv:2110.02905 [cs.LG].
R.Drautz,Phys.Rev.B 102,024104 (2020).
I. Kaliuzhnyi and C. Ortner, ArXiv e-prints 2202.04140 (2022).
A. Shapeev, Multiscale Model. Simul. 14, 1153 (2016).
J. P. Darby, D. P. Kovács, I. Batatia, M. A. Caro, G. L. W. Hart, C. Ortner, and G. Csányi, “Tensorreduced atomic density representations,” (2022).
I. M. Gelfand and M. L. Tsetlin, 825828, 71 (1950).
A. Alex, M. Kalus, A. Huckleberry, and J. von Delft, Journal of Mathematical Physics 52, 023507 (2011).
P.-Y. Lagrave, Y. Cabanes, and F. Barbaresco, in Geometric Science of Information, edited by F. Nielsen and F. Barbaresco (Springer International Publishing, Cham, 2021) pp. 577-584.
M. Favoni, A. Ipp, D. I. Müller, and D. Schuh, Phys. Rev. Lett. 128, 032003 (2022).
A. Bogatskiy, B. Anderson, J. Offermann, M. Roussi, D. Miller, and R. Kondor, in Proceedings of the 37th International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 119, edited by H. D. III and A. Singh (PMLR, 2020) pp. 992-1002.
A. Butter et al., SciPost Phys. 7, 014 (2019), arXiv:1902.09914 [hep-ph].
H. Qu, C. Li, and S. Qian, "Particle transformer for jet tagging," (2022).
H. Qu and L. Gouskos, Phys. Rev. D 101, 056019 (2020), arXiv:1902.08570 [hep-ph].
J. M. Munoz, I. Batatia, and C. Ortner, "Bip: Boost invariant polynomials for efficient jet tagging," (2022).
P. T. Komiske, E. M. Metodiev, and J. Thaler, JHEP 01, 121 (2019), arXiv:1810.05165 [hep-ph].
J. Pearkes, W. Fedorko, A. Lister, and C. Gay, (2017), arXiv:1704.02124 [hep-ex].
Z. Wu, S. Song, A. Khosla, F. Yu, L. Zhang, X. Tang, and J. Xiao, in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE Computer Society, Los Alamitos, CA, USA, 2015) pp. 1912-1920.
X. Yan, github.com/yanx27 (2019).
C. R. Qi, H. Su, K. Mo, and L. J. Guibas, arXiv preprint arXiv:1612.00593 (2016).
C. R. Qi, L. Yi, H. Su, and L. J. Guibas, arXiv preprint arXiv:1706.02413 (2017).
Y. Shen, C. Feng, Y. Yang, and D. Tian, in CVPR (2018).
J. Li, B. M. Chen, and G. H. Lee, CoRR (2018).
S. Kumawat and S. Raman, in CVPR (2019).
A. I. Molev, “A weight basis for representations of even orthogonal lie algebras,” (1999), arXiv:math/9902060 [math.RT].
R. Steinberg, Bull. Amer. Math. Soc. 67, 406 (1961).
M. Bachmayr, G. Dusson, C. Ortner, and J. Thomas, ArXiv e-prints 2109.14771 (2021).
J. Thomas, H. Chen, and C. Ortner, Arch. Ration. Mech. Anal. 246 (2022).
T. Broecker, Representation of Compact Lie Groups (Springer Berlin Heidelberg, 1985).
N. Bourbaki, Lie Groups and Lie Algebras (Springer Berlin, Heidelberg, 1989).
A.1 Background on Lie groups and Tensor Products of Representations
A.1.1 Lie groups
Lie groups generalize the idea of continuous symmetry such as the rotation symmetry of the round sphere  $S^n$  (given by the action of the orthogonal group  $\mathrm{O}(n)$ ), the symmetry of a vector space under changes of basis (given by the action of the general linear group  $\mathrm{GL}_n(\mathbb{R})$ ), or the invariance of Minkowski spacetime under boosts (given by the Lorentz group). The symmetry groups of specific geometries were studied in the 19th century, leading to Sophus Lie's study of general continuous symmetry. Other major contributors to our understanding include Klein, Riemann, Hilbert, Poincaré, Noether, Cartan, among many others. In particular the solution of Hilbert's 5th problem by Gleason, Montgomery, and Zippin shows that under very general conditions a continuous symmetry group must be a Lie group. Symmetry being central to mathematics Lie groups appear in many areas of mathematics, from differential geometry to number theory.
Let us give the general formal definition, as well as a more practical concrete one:
Definition A.1. A real (complex) Lie group is a group that is also a finite-dimensional smooth (complex) manifold in which the product and inversion of the group are also smooth (holomorphic) maps.
A linear Lie group is a subgroup of the group  $\mathrm{GL}_n(\mathbb{R})$  which is also a closed subset (in the sense of real analysis).
It is a fact that every closed subgroup of a Lie group is a Lie group, so every group of the second type is indeed also a group of the first type. Conversely while in general a Lie group is only locally isomorphic to a linear group (by a surjective group homomorphism), every compact Lie group is isomorphic to a linear group, and most Lie groups that arise in practice are linear.
When the Lie group  $G$  acts on a space  $X$  (e.g. as a symmetry group) it naturally also acts on spaces of functions on  $X$  (by translation), and this action is linear. In general, an action of  $G$  on a (suually complex) vector space by linear map is called a linear representation (of  $G$ ). In the context of our paper the space of functions on  $X$  is the space of features, and we achieve efficient computation in it by decomposing this representation into simpler constituents – in other words by understanding the representation theory of  $G$ .
Describing the representation theory of arbitrary Lie groups is a very difficult problem and an active area of research in pure mathematics. However, for certain classes of groups it is possible to say quite a bit (for example classical harmonic analysis can be viewed from the lens of the representation theory of commutative groups). An important class of groups whose representation theory can be completely understood is the class of reductive Lie groups, on which we focus in this paper.
For now let us give a concrete definition; an abstract will follow later once we introduce the necessary language.
Definition A.2. A linear Lie group  $G \subset \mathrm{GL}_n(\mathbb{R})$  is reductive if it is also closed under transpose: for every  $g \in G$  we have that  $g^T \in G$ .
Most of the groups that are of interest in natural sciences are reductive Lie groups. This includes well-known groups such as the symplectic group  $\mathrm{Sp}_{2n}(\mathbb{R})$  relevant in Hamiltonian dynamics, and orthogonal group  $\mathrm{O}(n)$  which parametrizes rotations in  $\mathbb{R}^n$ .
A.1.2 Linear representations
Fix a Lie group  $G$ . A linear representation of  $G$  is a pair  $(\pi, V)$  where  $V$  is a complex vector space and  $\pi$  is an action of  $G$  on  $V$  by linear maps. In other words for each  $g \in G$  we have a complex-linear map  $\pi(g) \colon V \to V$  such that  $\pi(g_1g_2) = \pi(g_1)\pi(g_2)$  and such that  $\pi(1_G) = \mathrm{Id}_V$  (in other words,  $\pi \colon G \to \mathrm{GL}(V)$  is a group homomorphism).