Homology Representations Arising from a Hypersimplex

TL;DR

This paper uses discrete Morse theory to analyze the homology of subcomplexes derived from hypersimplices, revealing symmetric group actions and character descriptions.

Contribution

It provides a complete acyclic matching for the face lattice of hypersimplices and classifies subcomplexes with homology concentrated in one degree.

Findings

Classified subcomplexes with homology in a single degree.

Described symmetric group actions on homology groups.

Provided character formulas for these actions.

Abstract

We present a complete acyclic matching of the Hasse diagram associated with the face lattice of a hypersimplex. Since a hypersimplex is a convex polytope, there is a natural way to form a CW complex from its faces. We will then utilize this matching along with discrete Morse theory and some topological techniques to classify every subcomplex whose reduced homology groups are concentrated in a single degree. These reduced homology groups support a natural action of the symmetric group and a description of the characters that this action produces is given.