On Twisted Tensor Product Group Embeddings and the Spin Representation of Symplectic Groups: The Case 𝑞 Odd

Abstract

The group 𝑃𝑆𝑝₈(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).

1. Introduction

Let 𝐺 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=𝑁_𝐺(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].

Let 𝐆 be a finite classical group with natural module 𝑉₀ of dimension 𝑛≥2 over the Galois field GF(𝑞^𝑡). Let 𝑉^𝜓𝑖₀ denote the 𝐆-module 𝑉₀ with group action given by 𝑔⋅𝑣=𝑔^𝜓𝑖(𝑣), where 𝑔^𝜓𝑖 denotes the matrix 𝑔 with its entries raised to the 𝑞^𝑖th power, 𝑖=0,…,𝑡−1. Then one can form the so-called twisted tensor product module 𝑉₀⊗𝑉^𝜓₀⊗⋯⊗𝑉^𝜓𝑡−1₀. Such a module can be realized over the subfield GF(𝑞) of GF(𝑞^𝑡). This gives rise to an absolutely irreducible representation of the group 𝐆 on an 𝑛^𝑡-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(𝑞^𝑡)≤𝑃Ω⁺_2𝑡(𝑞), where 𝑡≥2 and 𝑞 is even; and 𝑃𝑆𝑝_2𝑚(𝑞^𝑡)≤𝑃Ω^𝜖_(2𝑚)𝑡(𝑞) with 𝑞 even. We will use 𝑆𝑝₂(𝑞) although some references will use SL₂(𝑞). We found that our embedding of 𝑃𝑆𝑝₂(𝑞^𝑡) is associated with an embedding of the projective line 𝑃𝐺(1,𝑞^𝑡) as a complete partial ovoid of a quadric in 𝑃𝐺(2^𝑡−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 𝑃𝑆𝑝_2𝑚(𝑞^𝑡), 𝑞 even, in 𝑃Ω^𝜖_(2𝑚)𝑡(𝑞) has a particular application when 𝑚=2 in the embeddings of symplectic ovoids of 𝑃𝐺(3,𝑞^𝑡) as partial ovoids of hyperbolic quadrics of 𝑃𝐺(4^𝑡−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 𝑛^𝑡-dimensional module over GF(𝑞) for 𝐆 may be viewed projectively as a subspace of the projective space 𝑃𝐺(_𝑡^𝑛𝑡−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 𝑃𝑆𝑝_2𝑚(𝑞^𝑡), 𝑚≤2, 𝑞 even, an intermediate embedding of type 𝒞₃ occurs: 𝑃𝑆𝑝_2𝑚(𝑞^𝑡)<𝑃𝑆𝑝_2𝑚𝑡(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 𝑆𝑝_2𝑚𝑡(𝑞).

Note that class 𝒞₉ 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(𝑞^𝑡)≤𝑃Ω⁺_2𝑡(𝑞) and 𝑃𝑆𝑝₄(𝑞^𝑡)≤𝑃Ω^𝜖_4𝑡(𝑞) with 𝑞 even.

In this paper we consider the twisted tensor product embedding of 𝑃𝑆𝑝₂(𝑞^𝑡) inside 𝑃𝑆𝑝_2𝑡(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 case 𝑛=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].

Let 𝐸_𝑖, 1≤𝑖≤3, be 2-dimensional vector spaces over GF(𝑞³) and let 𝐸=𝐸₁⊕𝐸₂⊕𝐸₃. Suppose that for each 𝑖, 𝑒_𝑖1,𝑒_𝑖2 is a basis for 𝐸_𝑖 and suppose that 𝐻≤GL(𝐸₁). For 𝑣=∑_𝑗𝜆_𝑗𝑒_𝑖𝑗∈𝐸_𝑖 we write 𝑣^Ψ=∑_𝑗𝜆^𝑞_𝑗𝑒_𝑖+1𝑗∈𝐸_𝑖+1 (with 𝑖+1 interpreted modulo 3), and for ℎ∈𝐻 we write ℎ^Ψ for the matrix ℎ with every entry raised to the power 𝑞. Hence, to any 𝑣∈𝐸₁ there correspond “conjugate” vectors 𝑣^Ψi−1∈𝐸_𝑖 and 𝐻 acts on 𝐸_𝑖 via ℎ^Ψ𝑖−1(𝑣^Ψ𝑖−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,𝑣^Ψ,𝑣^Ψ2⟩. 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+𝑣^Ψ+𝑣^Ψ2, 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 𝑓_𝑖 on 𝐸_𝑖 given by 𝑓_𝑖(𝑢^Ψ𝑖−1,𝑤^Ψ𝑖−1)=𝑓₁(𝑢,𝑤)^𝑞𝑖−1 and an alternating form 𝑓 on 𝐸 in which 𝑓_|𝐸𝑖 is 𝑓_𝑖 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 of 𝐸, ⋀³(𝐸), an 𝐻-module of dimension 63 over GF(𝑞³). If 𝐴⊕𝐵 is any decomposition for 𝐸, then ³(𝐸)=_𝑖+𝑗=3^𝑖(𝐴)⊗^𝑗(𝐵).(2.1)

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,𝑣^Ψ,𝑣^Ψ2⟩ is mapped to the 1-dimensional subspace corresponding to 𝑣∧𝑣^Ψ∧𝑣^Ψ2∈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 to 𝒮 and its image in Δ∩𝒢₀. We have seen that these points in Δ∩𝒢₀ may be represented by 𝑣⊗𝑣^Ψ⊗𝑣^Ψ2 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 have 𝐻 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 Embedding 𝑃𝑆𝑝₂(𝑞³)<𝑃𝑆𝑝₈(𝑞), 𝑞 Odd

We consider a vector space 𝑉 of dimension 6 and the corresponding projective space Σ₀=𝑃𝐺(5,𝑞).

Let ℐ 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].

When 𝑞 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(𝑉)׳(𝑉)⟶⁶(𝑉)=GF𝑞.(3.1) 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 embedding 𝑃𝑆𝑝₂(𝑞³)<𝑃Ω⁺₈(𝑞), 𝑞≥4 even, an intermediate 𝒞₃-embedding occurs: 𝑃𝑆𝑝₂(𝑞³)<𝑃𝑆𝑝₆(𝑞)<𝑃Ω⁺₈(𝑞).

Moreover, 𝑁_𝑃Ω⁺_8(𝑞)(𝑃𝑆𝑝₂(𝑞³)) is the stabilizer of 𝒪 in 𝑃Ω⁺₈(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 𝑃Ω⁺₈(𝑞). See also [18, 19].

Now, we focus on the case 𝑞 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 𝑃Ω⁺₈(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 normalizer 𝑁 of 𝑃𝑆𝑝₂(𝑞³) in 𝑃𝑆𝑝₈(𝑞) stabilizes 𝒪.

Proof. Let 𝐹 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 𝐹^∼. Suppose that 𝐹^∼ 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^∼−1𝐹ℎ 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 case 𝑞 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 group 𝑁 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 that 𝑁 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 𝑆_𝑃 of 𝒮 on 𝑃. Moreover, Stab_𝑁(𝑃) acts transitively on 𝑆_𝑃⧵{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,𝜔^𝑞3), 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(𝜔^𝑎,𝜔^𝑎𝑞,𝜔^𝑎𝑞2,𝜔^𝑎𝑞3,𝜔^𝑎𝑞4,𝜔^𝑎𝑞5,𝜔^𝑏,𝜔^𝑏𝑞), 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 ovoid 𝒪 is complete.

Proof. From the previous proposition it follows that 𝒪 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 𝑆_𝑃, 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 𝒪_𝑖 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.