Composition factors of 2-parts spin representations of symmetric groups

Lucia Morotti

doi:10.5802/alco.137

Abstract

Given an odd prime $p$, we identify composition factors of the reduction modulo $p$ of spin irreducible representations of the covering groups of symmetric groups indexed by partitions with 2 parts and find some decomposition numbers.

Highlights

Let Sn be a double cover of the symmetric group Sn
Let F be a field of characteristic p = 2 and V be an irreducible representation of F Sn
If z acts as 1 V may be viewed as a representation of Sn, if on the other hand z acts as −1 we say that V is a spin representation of Sn

Summary

Introduction

Let Sn be a double cover of the symmetric group Sn. There exists z ∈ Sn central of order 2 such that the sequence. It is well known that in characteristic 0 (pairs of) irreducible spin representations of the symmetric groups are labeled by partitions with distinct parts, that is strict partitions. Is a complete set of non-isomorphic irreducible spin representations of Sn. For λ ∈ RPp(n) we define the supermodule D(λ) to be either D(λ, 0) or D(λ, +) ⊕ D(λ, −) (depending on the parity of n − hp (λ)). For irreducible representations of symmetric groups an exact description of the composition factors, and their multiplicities, of the reduction modulo p of the modules S(λ1,λ2) had been obtained by James in [11]. Theorem 1.3 gives new information about composition factors (and their multiplicities) of the reduction modulo p of the modules S((n − j, j)).

Basic lemmas

Projective modules