Projective representations of symmetric groups via Sergeev duality

Abstract

Listen

Translate article

In this article, we determine the irreducible projective representations of the symmetric group Sd and the alternating group Ad over an algebraically closed field of characteristic p 6= 2. These matters are well understood in the case p = 0, thanks to the fundamental work of Schur [24] in 1911, as well as the much more recent work of Nazarov [19, 20], Sergeev [25, 26] and others. So the focus here is primarily on the case of positive characteristic, where surprisingly little is known as yet. In particular, we obtain a natural combinatorial labelling of the irreducibles in terms of a certain set RPp(d) of restricted p-strict partitions of d. Such partitions arose recently in work of Kashiwara, Miwa, Peterson and Yung [11] and Leclerc and Thibon [14] on the q-deformed Fock space of the affine Kac-Moody algebra of type A p−1. Leclerc and Thibon proposed that RPp(d) should label the irreducible projective representations in some natural way, and we show here how this can be done. Note that for p = 3, 5, the labelling problem was solved in [1, 3], while if p = 2 all projective representations of Sd and Ad are linear so do not need to be considered further here. To be more precise, recall that λ is a partition of d if λ = (λ1, λ2, . . . ) is a non-increasing sequence of non-negative integers summing to d. Call λ p-strict if in addition