Annihilators of permutation modules

TL;DR

This paper provides an explicit combinatorial basis for the annihilator of permutation modules in symmetric group algebras, revealing its structure as a cell ideal in Murphy's framework, with implications for semisimple and non-semisimple cases.

Contribution

It introduces a new combinatorial basis for the annihilator of permutation modules, connecting it to Murphy's cell structure in Hecke algebras.

Findings

The annihilator forms a cell ideal in the semisimple case.

Explicit combinatorial basis constructed for the annihilator.

Results may not hold in non-semisimple cases.

Abstract

Permutation modules are fundamental in the representation theory of symmetric groups $\Sym_n$ and their corresponding Iwahori--Hecke algebras $\He = \He(\Sym_n)$. We find an explicit combinatorial basis for the annihilator of a permutation module in the "integral" case -- showing that it is a cell ideal in G.E. Murphy's cell structure of $\He$. The same result holds whenever $\He$ is semisimple, but may fail in the non-semisimple case.