We classify the small irreducible representations of Sp 2n (q) with q even in odd characteristic. This improves even the known results for complex representations. The smallest representation for this group is much larger than in the case when q is odd. This makes the problem much more difficult.

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Cross characteristic representations of even characteristic symplectic groups

We classify all pairs (G,V) with G a closed subgroup in a classical group with natural module V over the complex numbers such that G has the same composition factors on the kth tensor power of V, for a fixed (small) k. In particular, we prove Larsen's conjecture stating that for dim(V) > 6 and k = 4, there are no such G aside from those containing the derived subgroup of the classical group. We also find all the examples where this fails for dim(V) < 7. As a consequence of our results, we obtain a short proof of a related conjecture of Katz. These conjectures are used in Katz's recent works on monodromy groups attached to Lefschetz pencils and to character sums over finite fields. Modular versions of these conjectures are also studied, with a particular application to random generation in finite groups of Lie type.

Decompositions of small tensor powers and Larsen’s conjecture

We investigate the question when the tensor square, the alternating square, or the symmetric square of an absolutely irreducible projective representation V of an almost simple group G is again irreducible. The knowledge of such representations is of importance in the description of the maximal subgroups of simple classical groups of Lie type. We show that if G is of Lie type in odd characteristic, either V is a Weil representation of a symplectic or unitary group, or G is one of a finite number of exceptions. For G in even characteristic, we derive upper bounds for the dimension of V which are close to the minimal possible dimension of nontrivial irreducible representations. Our results are complete in the case of complex representations. We will also answer a question of B. H. Gross about finite subgroups of complex Lie groups & that act irreducibly on all fundamental representations of &.

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Irreducibility of tensor squares, symmetric squares and alternating squares

Decompositions of Small Tensor Powers and Larsen's Conjecture

Finite groups admitting grassmannian 4-designs☆

Let G be a nite quasi-simple group of Lie type deened over a eld of characteristic p. We determine whether G can possess irreducible Brauer characters , (of degree > 1) in characteristic other than p such that is irreducible.

Ρham Huu Tiep

Papers

Irreducible tensor products of representations of finite quasi-simple groups of Lie type