Irreducible representations of the symmetric groups from slash homologies of p-complexes

TL;DR

This paper computes slash homology of p-complexes related to symmetric groups, revealing irreducible representations linked to two-row partitions and introducing a basis via p-standard tableaux.

Contribution

It provides explicit calculations of slash homology for p-complexes of symmetric group representations, connecting homological invariants to irreducible modules and tableaux.

Findings

Slash homology groups correspond to irreducible two-row partition representations.

Every non-trivial slash homology appears as an irreducible symmetric group representation.

A basis for these representations is given by p-standard tableaux.

Abstract

In the 40s, Mayer introduced a construction of (simplicial) $p$-complex by using the unsigned boundary map and taking coefficients of chains modulo $p$. We look at such a $p$-complex associated to an $(n-1)$-simplex; in which case, this is also a $p$-complex of representations of the symmetric group of rank $n$ - specifically, of permutation modules associated to two-row compositions. In this article, we calculate the so-called slash homology - a homology theory introduced by Khovanov and Qi - of such a $p$-complex. We show that every non-trivial slash homology group appears as an irreducible representation associated to two-row partitions, and how this calculation leads to a basis of these irreducible representations given by the so-called $p$-standard tableaux.