On the classification of Specht modules with one-dimensional summands

Abstract

This paper extends a result of James to a combinatorial condition on partitions for the corresponding Specht module to have a summand isomorphic to the unique one-dimensional $F\mathrm{\Sigma }$-module over fields of characteristic 2. The work makes use of a recursively defined condition to reprove a result of Murphy and prove a new result for self-conjugate partitions. Finally we present a Python script which utilizes this work to test Specht modules for a one-dimensional summand.