Differential posets and restriction in critical groups

TL;DR

This paper explores the structure of critical groups associated with finite group representations, revealing connections to Cayley graphs, differential posets, and providing generalizations of known formulas.

Contribution

It introduces a framework linking critical groups of group representations to graph coverings and differential posets, extending previous results and formulas.

Findings

Critical groups of abelian groups are isomorphic to those of certain Cayley graphs.

Restriction maps between critical groups correspond to graph covering maps.

Generalization of explicit formulas for critical groups of symmetric group representations.

Abstract

In recent work, Benkart, Klivans, and Reiner defined the critical group of a faithful representation of a finite group $G$, which is analogous to the critical group of a graph. In this paper we study maps between critical groups induced by injective group homomorphisms and in particular the map induced by restriction of the representation to a subgroup. We show that in the abelian group case the critical groups are isomorphic to the critical groups of a certain Cayley graph and that the restriction map corresponds to a graph covering map. We also show that when $G$ is an element in a differential tower of groups, critical groups of certain representations are closely related to words of up-down maps in the associated differential poset. We use this to generalize an explicit formula for the critical group of the permutation representation of the symmetric group given by the second author, and to enumerate the factors in such critical groups.