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On twisted tensor product group embeddings and the spin representation of symplectic groups. (English) Zbl 1154.20039

Let \(G\) be a finite classical group with natural module \(V_0\) of dimension greater than or equal to \(2\) over the finite field \(\mathbb{F}_{q^t}\). For each \(0\leq i\leq t-1\), and each \(g\in G\), let \(g^{\psi^i}\) denote the matrix obtained from the matrix \(g\) by raising all entries to the power \(q^i\). Then we obtain \(G\)-modules \(V_0^{\psi^i}\) whose underlying space is \(V_0\) with the \(G\)-action twisted by \(\psi^i\). The tensor product module \(\bigotimes_{i=0}^{t-1}V_0^{\psi^i}\) can be realised over \(\mathbb{F}_q\), giving rise to an irreducible representation of \(G\) on an \(n^t\)-dimensional module over \(\mathbb{F}_q\), and hence a so-called ‘twisted tensor product group embedding’ of \(G\) in a classical group over \(\mathbb{F}_q\), [see R. Steinberg, Nagoya Math. J. 22, 33-56 (1963; Zbl 0271.20019)]. It was suggested by G. M. Seitz [Geom. Dedicata 25, 391-406 (1988; Zbl 0716.20009)] that the class of such subgroups should be labelled \(\mathcal C_9\), following the description of various classes subgroups of classical groups by M. Aschbacher [Invent. Math. 76, 469-514 (1984; Zbl 0537.20023)].
In this paper, the authors study these embeddings in a geometric context. In particular, they show that the \(n^t\)-dimensional \(G\)-module described above may be viewed as a subspace of the projective space \(\text{PG}({nt\choose t}-1,q)\) containing the Grassmannian of \((t-1)\)-subspaces of \(\text{PG}(nt-1,q)\). They apply this geometric point of view to some specific examples in low dimension, notably the embeddings \(\text{PSp}_{2m}(q^t)<\text{P}\Omega_{(2m)^t}(q)\) for \(m\leq 2\) and \(q\) even. Here they are able to show that the subgroup \(\text{PSp}_{2m}(q^t)\) is not maximal (an intermediate embedding of type \(\mathcal C_3\) occurs). This was already known to M. Schaffer [Commun. Algebra 27, No. 10, 5097-5166 (1999; Zbl 0938.20038)], but his proof uses the Classification of Finite Simple Groups, which is not required here.

MSC:

20G40 Linear algebraic groups over finite fields
51N30 Geometry of classical groups
20E07 Subgroup theorems; subgroup growth
20E28 Maximal subgroups
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