The multistep homology of the simplex and representations of symmetric groups

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.

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

“…sends a k-subset ω to the formal sum of all (k − a)subsets contained in ω. This map is called a-step boundary map in [19] and we will follow this convention. The multi-step boundary maps satisfy the following (super-)Leibniz rule, which is called splitting lemma in [19].…”