On restrictions of modular spin representations of symmetric and alternating groups

Abstract

Abstract. Let F be an algebraically closed field of characteristic p and H be an almost simple group or a central extension of an almost simple group. An important problem in representation theory is to classify the subgroups G of H and FH-modules V such that the restriction V ↓ G is irreducible. For example, this problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where H is the Schur's double coverÂn orŜn.

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“…The Main Theorem substantially strengthens Theorem A of [Kleshchev and Tiep 2004], which in turn strengthened [Wagner 1977], and fits naturally into the program of describing small dimension representations of quasisimple groups. For representations of symmetric and alternating groups results along these lines were obtained in [James 1983] and [Brundan and Kleshchev 2001b, Section 1].…”

“…Here we review some known results on representation theory ofn and ᐁ n described in detail in [Kleshchev 2005, Chapter 22] following [Brundan and Kleshchev 2001a;. It is important that the different approaches of these last two papers are reconciled in [Kleshchev and Shchigolev 2012], where some additional branching results, which will be crucial for us here, are also established.…”

Some remarks on maximal subgroups of finite classical groups

“…Factorizations of irreducible characters of alternating and symmetric groups and their Schur covers into Kronecker products of irreducible characters are very rare. The relevant references are Kleshchev and Bessenrodt [8], [9] and [10] as well as Kleshchev and Tiep [25]. It is an open question as to which of these factorizations do in fact lead to C 4obstructions.…”

“…Recent work of Kleshchev, Sin, Tiep [24] goes a long way towards solving the branching problems when = 2 or 3 and G = A n or S n . Kleshchev and Tiep [25] nearly obtained a complete solution in the case G = 2.A n or 2.S n and < n.…”

Irreducible tensor products of representations of covering groups of symmetric and alternating groups

“…For alternating groups, apart for some cases in in characteristic 2, non-trivial tensor products have been classified in [5,7,37,38,46]. For covering groups of symmetric and alternating groups however only partial results are known, that is the characteristic 0 case for S n , see [4,8], as well as some reduction results obtained in [34] for S n and A n in characteristic ≥ 5. In this paper we will consider the case where G = S n or A n is a covering group of a symmetric or alternating group and completely classify non-trivial irreducible tensor products in characteristic = 2.…”