Fixed-point functors for symmetric groups and Schur algebras

Abstract

Let Σ d be the symmetric group. For 1< m< d let F m be the functor which takes a Σ d -module U to the space of fixed points U Σ m , which is naturally a module for Σ d− m . This functor was previously used by the author to study cohomology of the symmetric group, but little is known about it. This paper initiates a study of F m . First, we relate it to James' work on row and column removal and decomposition numbers for the Schur algebra. Next, we determine the image of dual Specht modules, permutation and twisted permutation modules, and some Young and twisted Young modules under F m . In particular, F m acts as first row removal on dual Specht modules S λ with λ 1= m and as first column removal on twisted Young and twisted permutation modules corresponding to partitions with m parts. Finally, we prove that determining F m on the Young modules is equivalent to determining the decomposition numbers for the Schur algebra.