Irreducibility of tensor squares, symmetric squares and alternating squares

Abstract

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 G that act irreducibly on all fundamental representations of G.

Primes dividing the degrees of the real characters

“…By [1, Lemma 3.1], µ belongs to the principal r -block of S. Viewed as a character of H * , µ also belongs to the principal r -block of H * . Also, since µ(1) is coprime to p, µ is semisimple by [21,Lemma 7.2]. It now follows by the fundamental result of Broué and Michel [3] that µ is the semisimple character…”

Finite groups admitting grassmannian 4-designs

“…Our further analysis relies on the following upper bound on the dimension of V , cf. also [MMT,GT2]. Then d = dim(V ) < 3/2 + √ 2(G : N).…”

“…[MMT,Corollary 2.10] Let S be a finite simple group of Lie type and Z a long-root subgroup ofŜ. Then either (5) holds, or S ∈ PSL 2 (5), SU 3 (3), SU 4 (2) .…”

“…Assume L = 3 · S or 6 · S. Then V is not self-dual. Arguing as in [MMT,p. 391], we see that dim (V ) √ 6(G : C G (Z)) + 1 < 4877, which is a contradiction by [Lu].…”

“…Even though our condition on V is in general weaker than the one considered in [GT2], one can still establish the crucial reduction to the two cases, extraspecial and almost quasisimple. In fact, we will use some results from [GT2,LST,MMT], and we will take this opportunity to correct an inaccuracy in [GT2], cf. Section 5.3 below.…”