Character values and decomposition matrices of symmetric groups

Abstract

By exploiting relationships between the values taken by ordinary characters of symmetric groups we prove two theorems in the modular representation theory of the symmetric group. 1. The decomposition matrices of symmetric groups in odd characteristic have distinct rows. In characteristic 2 the rows of a decomposition matrix labelled by the different partitions $λ$ and $μ$ are equal if and only if $λ$ and $μ$ are conjugate. An analogous result is proved for Hecke algebras. 2. A Specht module for the symmetric group $S_n$, defined over an algebraically closed field of odd characteristic, is decomposable on restriction to the alternating group $A_n$ if and only if it is simple, and the labelling partition is self-conjugate. This result is generalised to an arbitrary field of odd characteristic.