# On Twisted Tensor Product Group Embeddings and the Spin

Authors: Antonio Cossidente
Journal: ISRN Geometry
 (2011)

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## Abstract

The groupPSp8(q),q odd, has a maximal subgroup isomorphic to
3.PSp2(q3) belonging to the Aschbacher class
𝒞9. It is the full stabilizer of a complete
partial ovoid and of a complete partial 3-spread of 𝒲7(q).


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## Body

## 1. Introduction

LetG be a classical group associated with a finite
dimensional vector space over GF(q), say V. In his celebrated paper [1], Aschbacher describes a family 𝒞 of eight “geometric” classes of
subgroups of G and shows that any subgroup of
G either lies in one of these classes or has the form
H=NG(X), for some quasisimple subgroup
X of G satisfying some special conditions. Given such a
group H not lying in one of the eight classes of
𝒞, the main purpose is to determine whether or not
H is maximal in G. If not, there exists a quasisimple subgroup
K with X<K<G and one wants to study such configurations,
possibly from a geometric viewpoint. For more details, see [2].LetG be a finite classical group with natural module
V0 of dimension n≥2 over the Galois field GF(qt). Let V0ψi denote the G-module V0 with group action given by
g·v=gψi(v), where gψi denotes the matrix g with its entries raised to the
qith power, i=0,…,t-1. Then one can form the so-called twisted
tensor product moduleV0⊗V0ψ⊗⋯⊗V0ψt-1. Such a module can be realized over the subfield
GF(q) of GF(qt). This gives rise to an absolutely irreducible
representation of the group G on an nt-dimensional natural module over
GF(q). If G is a symplectic group, then under the twisted
tensor product embedding G turns out to be again a subgroup of a symplectic
group, and only when q is even, it is actually a subgroup of an orthogonal
group, see [3].Such representations are given by Steinberg [4] and further studied by Seitz [2].
See also [5]. We refer to this class of
subgroups as 𝒞9, as suggested by Seitz.In [6] we studied the geometry of two classes
of twisted tensor product group embeddings: PSp2(qt)≤PΩ2t+(q), where t≥2 and q is even; and PSp2m(qt)≤PΩ(2m)tϵ(q) with q even. We will use Sp2(q) although some references will use
SL2(q). We found that our embedding of
PSp2(qt) is associated with an embedding of the projective
line PG(1,qt) as a complete partial ovoid of a quadric in
PG(2t-1,q) (i.e., a maximal set of pairwise nonorthogonal
points of the quadric); if t≥3, then the quadric is hyperbolic. Such partial
ovoids are of some interest because their size attains the Blokhuis-Moorhouse bound
[7]. In particular, when
t=3 and q≥4, the embedding yields another description of the
Desarguesian ovoid of the hyperbolic quadric of PG(7,q) [8].
Similarly, the embedding of PSp2m(qt), q even, in PΩ(2m)tϵ(q) has a particular application when
m=2 in the embeddings of symplectic ovoids of
PG(3,qt) as partial ovoids of hyperbolic quadrics of
PG(4t-1,q) again with the size attaining the
Blokhuis-Moorhouse bound.In [9] we investigated further these twisted
tensor product group embeddings, but from a different perspective. We showed how the
nt-dimensional module over GF(q) for G may be viewed projectively as a subspace of the
projective space PG((ntt)-1,q) containing the Grassmannian of
(t-1)-subspaces of PG(nt-1,q). From this viewpoint G preserves the intersection of the subspace and the
Grassmannian. When n=2m≤4, this approach enabled us to address some questions
on maximality. We proved that under the twisted tensor product group embedding of
PSp2m(qt), m≤2, q even, an intermediate embedding of type
𝒞3 occurs: PSp2m(qt)<PSp2mt(q)<PΩ(2m)t(q). The partial ovoid referred to above lies on a
unique quadric in PG((2m)t-1,q). It turns out that this quadric is precisely that
arising from the spin representation of Sp2mt(q).Note that class𝒞9 has also been studied by Schaffer in [10], where he used representation theory
techniques; his arguments rely on the Classification of Finite Simple Groups. He
eliminated a number of possibilities, largely when t is composite and showed that the remaining
subgroups in this class are maximal except in a small number of cases. The main
exceptions are precisely PSp2(qt)≤PΩ2t+(q) and PSp4(qt)≤PΩ4tϵ(q) with q even.In this paper we consider the twisted tensor product embedding ofPSp2(qt) inside PSp2t(q) when q is odd, in the smallest case, that is,
t=3. The normalizer N of PSp2(q3) in PSp8(q), that has structure 3.PSp2(q3), is maximal in PSp8(q) [10]. We
study the action of N on points of PG(7,q). It turns out that N is the full stabilizer in PSp8(q) of a complete partial ovoid and also of a complete
partial spread of the symplectic space 𝒲7(q). The partial ovoid is of some interest because of
its connections with the generalized hexagons of type 3D4(q) and G2(q), see [11].


## 2. The Geometric Approach to the Twisted Tensor Product Embedding

In this section, specializing to the casen=2 and t=3, we recall the alternative perspective for at least
some of the subgroups in the Aschbacher's class 𝒞9 given in [9].LetEi, 1≤i≤3, be 2-dimensional vector spaces over
GF(q3) and let E=E1⊕E2⊕E3. Suppose that for each i, ei1,ei2 is a basis for Ei and suppose that H≤GL(E1). For v=∑jλjeij∈Ei we write vΨ=∑jλjqei+1j∈Ei+1 (with i+1 interpreted modulo 3), and for
h∈H we write hΨ for the matrix h with every entry raised to the power
q. Hence, to any v∈E1 there correspond “conjugate” vectors
vΨi-1∈Ei and H acts on Ei via hΨi-1(vΨi-1)=(hv)Ψi-1. Therefore we have an action of
H on E and H preserves a fibration of E into 3-dimensional subspaces of the form
〈v,vΨ,vΨ2〉. In projective terms, E corresponds to a projective space
Σ=PG(5,q3) and H preserves a partial 2-spread of
Σ. We may regard Ψ as a semilinear map on E. The vectors in E fixed by Ψ are precisely the vectors v+vΨ+vΨ2, where v∈E1, and they form a 6-dimensional vector space
V over GF(q) that spans E and is preserved by H. In Σ we have a set of points preserved by
H forming a subgeometry Σ0=PG(5,q) and on restriction, the partial 2-spread above
becomes a 2-spread 𝒮 of Σ0 preserved by H. Suppose that H preserves a nondegenerate alternating form
f1 on E1, then H preserves the alternating form
fi on Ei given by fi(uΨi-1,wΨi-1)=f1(u,w)qi-1 and an alternating form f on E in which f|Ei is fi and in which E1⊕E2⊕E3 is an orthogonal decomposition. Moreover the
restriction of f to V is a nondegenerate alternating form on
V. Thus H acts as a subgroup of Sp2(q3) embedded in Sp6(q) on Σ0 preserving a spread 𝒮 consisting now of totally isotropic planes.Consider the 3-fold alternating product ofE, ⋀3(E), an H-module of dimension (63) over GF(q3). If A⊕B is any decomposition for E, then (2.1)⋀3(E)=⨁i+j=3(⋀i(A)⊗⋀j(B)).Thus⋀3(E) has a subspace ⋀1(E1)⊗⋀2(E2⊕E3) and, by iteration, a subspace
⋀1(E1)⊗⋀1(E2)⊗⋀1(E3), that is, E1⊗E2⊗E3. This latter subspace is preserved by
H. The 3-dimensional subspaces of
E correspond to 1-dimensional subspaces of
⋀3(E). Each 3-dimensional GF(q)-subspace of V determines a 3-dimensional
GF(q3)-subspace of E and so ⋀3(V) may be regarded as a GF(q)-subspace of ⋀3(E). For any v∈E1, the 3-subspace 〈v,vΨ,vΨ2〉 is mapped to the 1-dimensional subspace
corresponding to v∧vΨ∧vΨ2∈E1⊗E2⊗E3. In projective terms PG((63)-1,q3) contains the Grassmannian 𝒢 of planes of Σ and E1⊗E2⊗E3 corresponds to a 7-dimensional subspace
Δ of PG((63)-1,q3) containing the image of the partial spread and it
is fixed by H. The planes of Σ0 form a Grassmannian 𝒢0 lying in a projective space
PG((63)-1,q) that is a subgeometry of PG((63)-1,q3). Each of the subspaces of 𝒮 is mapped into Δ∩𝒢0.As showed in [5, 2.4.1] the points of
PG(1,q3) may be represented as points of
PG(7,q). Given that G preserves the set of all such points and that
G acts irreducibly, these points must span
PG(7,q).Let us return to𝒮 and its image in Δ∩𝒢0. We have seen that these points in
Δ∩𝒢0 may be represented by v⊗vΨ⊗vΨ2 as v varies in E1. Moreover we may take H to be the group SL2(q3) acting absolutely irreducibly on
E1. Hence the points corresponding to
𝒮 generate a GF(q)-subspace Δ0 of projective dimension 5. It follows that the
GF(q3)-span of 𝒮 is precisely Δ. Hence we see the twisted tensor product module for
SL2(q3) as the subspace Δ0 of PG((63)-1,q).Observe that in one setting we haveH acting as a subgroup of GL2(q3) on PG(5,q), so here it is an Aschbacher
𝒞3 group. In a second setting it is a subgroup of
GL8(q) and lies in Aschbacher class
𝒞9.


## 3. The EmbeddingPSp2(q3)<PSp8(q), q Odd

We consider a vector spaceV of dimension 6 and the corresponding projective
space Σ0=PG(5,q).Letℐ be the set of all totally isotropic planes of
Σ0 with respect to a nondegenerate alternating form
f and let 𝒮 be a regular spread of Σ0 (with elements in ℐ). Then the Grasmannian, 𝒢0, of planes of Σ0 has dimension (63)-1 and the image of ℐ in 𝒢0 spans a subspace F3 of dimension (63)-(61)-1. The vector space equivalent of
F3 is the Weyl module of Sp6(q) for the fundamental weight
λ3. When q is even, F3 has a unique maximal subspace fixed by
PSp6(q), denoted N3. The quotient space M3=F3/N3 has dimension 7 and corresponds to the spin module
for Sp6(q). For more details, see [3, 12–16].Whenq is odd, M3 is the direct sum of the three twists of
E1 and their twisted tensor product. The symplectic
form on ⋀3(V) is given by the wedge product (3.1)⋀3(V)×⋀3(V)⟶⋀6(V)=GF(q). The restriction of this alternating form to
F3 must be nonsingular since F3 is a simple module. By projection, we get an
embedding of PSp2(q3) in PSp8(q), that is, PSp2(q3) in its twisted tensor product group
representation.In [9] we proved the following theorem.Theorem 3.1.
Under the twisted tensor product group embeddingPSp2(q3)<PΩ8+(q), q≥4 even, an intermediate 𝒞3-embedding occurs: PSp2(q3)<PSp6(q)<PΩ8+(q).Moreover,NPΩ8+(q)(PSp2(q3)) is the stabilizer of 𝒪 in PΩ8+(q) and it is a maximal subgroup of
PSp6(q).Remark 3.2.
It is a consequence of [17,
Theorem  I] that Sp6(q), q even, in its spin representation, is a maximal
subgroup of PΩ8+(q). See also [18, 19].Now, we focus on the caseq odd.It is easy to see that under the twisted tensor product embedding,PSp2(q3), q odd, turns out to be a subgroup of
PSp8(q) rather than a subgroup of PΩ8+(q), and it fixes a partial ovoid
𝒪 of 𝒲7(q) of size q3+1, that is, a set of q3+1 points no two of them conjugate with respect to
f.Lemma 3.3.
The normalizerN of PSp2(q3) in PSp8(q) stabilizes 𝒪.Proof.
LetF be the stabilizer in PSp2(q3) of a point of the projective line
PG(1,q3). Then F can also be described as the normalizer in
PSp2(q3) of a Sylow p-subgroup S of PSp2(q3) and there is a cyclic subgroup
C of F such that C∩S=1 and F=CS. Exactly one point of PG(1,q3) is fixed by F and exactly one point of
𝒪 is fixed by the image of
F under the twisted tensor product embedding, say
F̃. Suppose that F̃ fixes exactly one point
P of PG(7,q) (necessarily P will be in 𝒪). If h∈N, then h-1Fh fixes Ph, but h-1F̃h is the normalizer in PSL2(q3) of a Sylow p-subgroup so it fixes a point of
𝒪. Hence Ph∈𝒪. It follows that 𝒪h=𝒪.As in the caseq even, the normalizer N of PSp2(q3) in PSp8(q) has structure 3.PSp2(q3). It should be noted that N stabilizes also a partial spread of
𝒲7(q) of size q3+1 consisting of maximal totally isotropic subspaces
of 𝒲7(q) tangent to 𝒪. The action of N on 𝒪 and on 𝒮 is 2-transitive, see [11, Lemma  4.4(a)].Proposition 3.4.
The groupN has four orbits on points of
PG(7,q): 𝒪 of size q3+1, 𝒪1 of size q3(q3+1)(q-1)/2 consisting of points on secant lines to
𝒪; 𝒪2 of size (q3+1)(q3+q2+q) consisting of points on members of the partial
spread 𝒮 and 𝒪3 of size q3(q3-1)(q+1)/2.Proof.
It is sufficient to prove thatN has three orbits on PG(7,q)∖𝒪. Take 2 points P1 and P2 on 𝒪. The line L joining P1 and P2 is hyperbolic. The stabilizer of
P1,P2 in N acts transitively on L∖{P1,P2}. Since N acts 2-transitively on
𝒪, we get the orbit 𝒪1 of size q3(q3+1)(q-1)/2. If P∈𝒪, the stabilizer of P in N has order 3(q-1)q3(q2+q+1)/2. As we have seen, there is a unique member
SP of 𝒮 on P. Moreover, StabN(P) acts transitively on SP∖{P}. This way, we obtain the orbit
𝒪2 of size (q3+1)(q3+q2+q). To determine the fourth
N-orbit 𝒪3 we need some information on the twisted tensor
product embedding of a Singer cyclic group S of PSp(2,q3). In PSp2(q6) a Singer cycle has the diagonal representation
diag(ω,ωq3), where ω is a primitive element of
GF(q6) over GF(q). The (q+1)th power T of the twisted tensor product embedding of
S has the diagonal representation
diag(ωa,ωaq,ωaq2,ωaq3,ωaq4,ωaq5,ωb,ωbq), where a=q3+2q2+2q+1 and b=q5+q4+q3+q2+q+1. It turns out that T fixes a line ℓ pointwise and a projective 5-space
X setwise inducing a unitary Singer cyclic group
of order q2-q+1. In particular, from the diagonal
representation of T, we see that each T-orbit not on ℓ or X (that has size q2-q+1, generates a projective 6-space and its
stabilizer in N has order 3(q2-q+1). This way we get the N-orbit 𝒪3 of size q3(q3-1)(q+1)/2.Proposition 3.5.
The partial ovoid𝒪 is complete.Proof.
From the previous proposition it follows that𝒪 is a complete partial ovoid of
𝒲7(q). Indeed, if ℋ is a hyperplane of PG(7,q) then ℋ=𝒫⊥, for some point P, where ⊥ denotes the symplectic polarity. There exist
pairs P∈𝒪, Q∈𝒪1 with PQ totally isotropic, P∈𝒪, R∈𝒪2 with PR totally isotropic, P∈𝒪, T∈𝒪3 with PT totally isotropic. For, if
P∈𝒪, then P⊥ is a hyperplane of PG(7,q) which must meet any secant line
L to 𝒪, necessarily at a point
Q of 𝒪1, if L is not on P; and it contains SP, and so a point R of 𝒪2; finally, the remaining points of
P⊥ (which must exist, by counting) lie in
𝒪3. Hence, since 𝒪1,𝒪2,𝒪3 are orbits of N, each point of 𝒪i is collinear with a point of
𝒪 (in a totally isotropic line) and so cannot be
added to 𝒪 to obtain a partial ovoid. Thus
𝒪 is complete.



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*Source: 694605-2011-06-16.xml*