The multistep homology of the simplex and representations of symmetric groups

TL;DR

This paper investigates the homological properties of permutation modules of symmetric groups over fields of characteristic two, providing conditions for exactness and a new construction of basic spin modules.

Contribution

It introduces generalized boundary maps for permutation modules and characterizes their exactness, also offering an explicit construction of basic spin modules.

Findings

Determines when the chain complexes are exact or split exact.

Provides explicit construction of basic spin modules.

Generalizes boundary maps in simplicial homology.

Abstract

The symmetric group on a set acts transitively on its subsets of a given size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.