Differential posets and restriction in critical groups

Ayush Agarwal,Christian Gaetz

doi:10.5802/alco.70

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.

Highlights

The critical group K(Γ) is a well-studied abelian group invariant of a finite graph Γ which encodes information about the dynamics of a process called chip firing on the graph
This paper investigates maps on critical groups of group representations which are induced by group homomorphisms(1)
We will be interested in critical groups of graphs primarily as motivation for our study of critical groups of group representations, Section 3.1 below gives a close relationship between the two concepts when G is abelian

Summary

Introduction

The critical group K(Γ) is a well-studied abelian group invariant of a finite graph Γ which encodes information about the dynamics of a process called chip firing on the graph (see [7] where critical groups are called sandpile groups). Recent work of Benkart, Klivans, and Reiner defined analogous abelian group invariants K(V ), called critical groups, associated to a faithful representation V of a finite group G [2] It is known (see, for example, [18]) that graph covering maps induce surjective maps between graph critical groups. In [10], Miller and Reiner introduced a very strong conjecture about the Smith normal form of U D + tI where U, D are the up and down maps in a differential poset, and t is a variable We investigate how this conjecture, which was proven for powers of Young’s lattice by Shah [14], can be used to determine the structure of critical groups in certain differential towers of groups.

Background and definitions

Maps induced between critical groups

Critical groups and differential posets

Enumeration of factors in critical groups