ISRN GeometryVolume 2011 (2011), Article ID 694605, 6 pagesdoi:10.5402/2011/694605

Research Article

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

Antonio Cossidente

Dipartimento di Matematica e Informatica, Università degli studi della Basilicata, Contrada Macchia Romana, 85100 Potenza, Italy

Received 29 March 2011; Accepted 2 May 2011

Academic Editors: M. Khalkhali and U. Lindström

Copyright © 2011 Antonio Cossidente. Antonio Cossidente. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The group odd, has a maximal subgroup isomorphic to belonging to the Aschbacher class . It is the full stabilizer of a complete partial ovoid and of a complete partial 3-spread of .

1. Introduction

Let be a classical group associated with a finite dimensional vector space over , say . In his celebrated paper [1], Aschbacher describes a family of eight “geometric” classes of subgroups of and shows that any subgroup of either lies in one of these classes or has the form , for some quasisimple subgroup of satisfying some special conditions. Given such a group not lying in one of the eight classes of , the main purpose is to determine whether or not is maximal in . If not, there exists a quasisimple subgroup with and one wants to study such configurations, possibly from a geometric viewpoint. For more details, see [2].

Let be a finite classical group with natural module of dimension over the Galois field . Let denote the -module with group action given by , where denotes the matrix with its entries raised to the th power, . Then one can form the so-called twisted tensor product module . Such a module can be realized over the subfield of . This gives rise to an absolutely irreducible representation of the group on an -dimensional natural module over . If is a symplectic group, then under the twisted tensor product embedding turns out to be again a subgroup of a symplectic group, and only when 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 , as suggested by Seitz.

In [6] we studied the geometry of two classes of twisted tensor product group embeddings: , where and is even; and with even. We will use although some references will use . We found that our embedding of is associated with an embedding of the projective line as a complete partial ovoid of a quadric in (i.e., a maximal set of pairwise nonorthogonal points of the quadric); if , then the quadric is hyperbolic. Such partial ovoids are of some interest because their size attains the Blokhuis-Moorhouse bound [7]. In particular, when and , the embedding yields another description of the Desarguesian ovoid of the hyperbolic quadric of [8]. Similarly, the embedding of , even, in has a particular application when in the embeddings of symplectic ovoids of as partial ovoids of hyperbolic quadrics of 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 -dimensional module over for may be viewed projectively as a subspace of the projective space containing the Grassmannian of -subspaces of . From this viewpoint preserves the intersection of the subspace and the Grassmannian. When , this approach enabled us to address some questions on maximality. We proved that under the twisted tensor product group embedding of , , even, an intermediate embedding of type occurs: . The partial ovoid referred to above lies on a unique quadric in . It turns out that this quadric is precisely that arising from the spin representation of .

Note that class 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 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 and with even.

In this paper we consider the twisted tensor product embedding of inside when is odd, in the smallest case, that is, . The normalizer of in , that has structure , is maximal in [10]. We study the action of on points of . It turns out that is the full stabilizer in of a complete partial ovoid and also of a complete partial spread of the symplectic space . The partial ovoid is of some interest because of its connections with the generalized hexagons of type and , see [11].

2. The Geometric Approach to the Twisted Tensor Product Embedding

In this section, specializing to the case and , we recall the alternative perspective for at least some of the subgroups in the Aschbacher's class given in [9].

Let , , be 2-dimensional vector spaces over and let . Suppose that for each , is a basis for and suppose that . For we write (with interpreted modulo 3), and for we write for the matrix with every entry raised to the power . Hence, to any there correspond “conjugate” vectors and acts on via . Therefore we have an action of on and preserves a fibration of into 3-dimensional subspaces of the form . In projective terms, corresponds to a projective space and preserves a partial 2-spread of . We may regard as a semilinear map on . The vectors in fixed by are precisely the vectors , where , and they form a 6-dimensional vector space over that spans and is preserved by . In we have a set of points preserved by forming a subgeometry and on restriction, the partial 2-spread above becomes a 2-spread of preserved by . Suppose that preserves a nondegenerate alternating form on , then preserves the alternating form on given by and an alternating form on in which is and in which is an orthogonal decomposition. Moreover the restriction of to is a nondegenerate alternating form on . Thus acts as a subgroup of embedded in on preserving a spread consisting now of totally isotropic planes.

Consider the 3-fold alternating product of , , an -module of dimension over . If is any decomposition for , then

Thus has a subspace and, by iteration, a subspace , that is, . This latter subspace is preserved by . The 3-dimensional subspaces of correspond to 1-dimensional subspaces of . Each 3-dimensional -subspace of determines a 3-dimensional -subspace of and so may be regarded as a -subspace of . For any , the 3-subspace is mapped to the 1-dimensional subspace corresponding to . In projective terms contains the Grassmannian of planes of and corresponds to a 7-dimensional subspace of containing the image of the partial spread and it is fixed by . The planes of form a Grassmannian lying in a projective space that is a subgeometry of . Each of the subspaces of is mapped into .

As showed in [5, 2.4.1] the points of may be represented as points of . Given that preserves the set of all such points and that acts irreducibly, these points must span .

Let us return to and its image in . We have seen that these points in may be represented by as varies in . Moreover we may take to be the group acting absolutely irreducibly on . Hence the points corresponding to generate a -subspace of projective dimension 5. It follows that the -span of is precisely . Hence we see the twisted tensor product module for as the subspace of .

Observe that in one setting we have acting as a subgroup of on , so here it is an Aschbacher group. In a second setting it is a subgroup of and lies in Aschbacher class .

3. The Embedding , Odd

We consider a vector space of dimension 6 and the corresponding projective space .

Let be the set of all totally isotropic planes of with respect to a nondegenerate alternating form and let be a regular spread of (with elements in ). Then the Grasmannian, , of planes of has dimension and the image of in spans a subspace of dimension . The vector space equivalent of is the Weyl module of for the fundamental weight . When is even, has a unique maximal subspace fixed by , denoted . The quotient space has dimension 7 and corresponds to the spin module for . For more details, see [3, 12–16].

When is odd, is the direct sum of the three twists of and their twisted tensor product. The symplectic form on is given by the wedge product The restriction of this alternating form to must be nonsingular since is a simple module. By projection, we get an embedding of in , that is, in its twisted tensor product group representation.

In [9] we proved the following theorem.

Theorem 3.1. Under the twisted tensor product group embedding , even, an intermediate -embedding occurs: .

Moreover, is the stabilizer of in and it is a maximal subgroup of .

Remark 3.2. It is a consequence of [17, Theorem  I] that , even, in its spin representation, is a maximal subgroup of . See also [18, 19].

Now, we focus on the case odd.

It is easy to see that under the twisted tensor product embedding, , odd, turns out to be a subgroup of rather than a subgroup of , and it fixes a partial ovoid of of size , that is, a set of points no two of them conjugate with respect to .

Lemma 3.3. The normalizer of in stabilizes .

Proof. Let be the stabilizer in of a point of the projective line . Then can also be described as the normalizer in of a Sylow -subgroup of and there is a cyclic subgroup of such that and . Exactly one point of is fixed by and exactly one point of is fixed by the image of under the twisted tensor product embedding, say . Suppose that fixes exactly one point of (necessarily will be in ). If , then fixes , but is the normalizer in of a Sylow -subgroup so it fixes a point of . Hence . It follows that .

As in the case even, the normalizer of in has structure . It should be noted that stabilizes also a partial spread of of size consisting of maximal totally isotropic subspaces of tangent to . The action of on and on is 2-transitive, see [11, Lemma  4.4(a)].

Proposition 3.4. The group has four orbits on points of : of size , of size consisting of points on secant lines to ; of size consisting of points on members of the partial spread and of size .

Proof. It is sufficient to prove that has three orbits on . Take 2 points and on . The line joining and is hyperbolic. The stabilizer of in acts transitively on . Since acts 2-transitively on , we get the orbit of size . If , the stabilizer of in has order . As we have seen, there is a unique member of on . Moreover, Stab acts transitively on . This way, we obtain the orbit of size . To determine the fourth -orbit we need some information on the twisted tensor product embedding of a Singer cyclic group of . In a Singer cycle has the diagonal representation diag, where is a primitive element of over . The th power of the twisted tensor product embedding of has the diagonal representation diag), where and . It turns out that fixes a line pointwise and a projective 5-space setwise inducing a unitary Singer cyclic group of order . In particular, from the diagonal representation of , we see that each -orbit not on or (that has size , generates a projective 6-space and its stabilizer in has order . This way we get the -orbit of size .

Proposition 3.5. The partial ovoid is complete.

Proof. From the previous proposition it follows that is a complete partial ovoid of . Indeed, if is a hyperplane of then , for some point , where denotes the symplectic polarity. There exist pairs , with totally isotropic, , with totally isotropic, , with totally isotropic. For, if , then is a hyperplane of which must meet any secant line to , necessarily at a point of , if is not on ; and it contains , and so a point of ; finally, the remaining points of (which must exist, by counting) lie in . Hence, since are orbits of , each point of is collinear with a point of (in a totally isotropic line) and so cannot be added to to obtain a partial ovoid. Thus is complete.

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