New columns in decomposition matrices of symmetric groups for every block

Abstract

The central unsolved problem in the modular representation theory of symmetric groups is to find the decomposition matrices, which describe how irreducible representations in characteristic zero decompose upon reduction modulo a prime characteristic $p$. In this paper we determine a large number of new columns in these decomposition matrices, namely those labeled by partitions whose $p$-divisible hooks have all even arm lengths. In particular in odd characteristic $p$, for every possible block of every possible symmetric group $S_n$, we determine at least one complete column. These columns are multiplicity-free and are described by a recently introduced combinatorial statistic of partitions (depending on $p$), called the odd sequence. As an application, we determine the indecomposable summands of Foulkes modules $H^{(2^m)}$.