Speaker: Sheila Sundaram, Pierrepont School

Title: Plethysm and characters of $S_n$ induced from centralisers

Abstract: We study characters of the symmetric group $S_n$ induced from centraliser subgroups. There are interesting plethystic relationships among these characters, notably between the Lie character and the conjugacy action of $S_n.$

A classical result of Thrall states that the regular representation of the symmetric group $S_n$ decomposes into a sum of symmetric powers of the representation afforded by the Lie character. By contrast, here we arrive at a decomposition of the regular representation as a sum of exterior powers of modules related to the Lie representation of $S_n.$

We also investigate the Schur-positivity of multiplicity-free sums of power-sums, and give a general plethystic formula for such sums.