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

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 two-row partitions, and how this calculation leads to a basis of these irreduicble representations given by the so-called p-standard tableaux.