The absolute order of a permutation representation of a Coxeter group

TL;DR

This paper introduces and studies a family of partial orders derived from permutation representations of Coxeter groups, exploring their properties, divisibility conditions, and applications to subgroups like the alternating group.

Contribution

It defines a new family of absolute orders from permutation representations and investigates their algebraic properties and applications, including subgroup structures.

Findings

Conditions for divisibility of rank generating polynomials are established.

Examples include symmetric group actions on perfect matchings.

A well-behaved absolute order on the alternating subgroup is constructed.

Abstract

A permutation representation of a Coxeter group $W$ naturally defines an absolute order. This family of partial orders (which includes the absolute order on $W$) is introduced and studied in this paper. Conditions under which the associated rank generating polynomial divides the rank generating polynomial of the absolute order on $W$ are investigated when $W$ is finite. Several examples, including a symmetric group action on perfect matchings, are discussed. As an application, a well-behaved absolute order on the alternating subgroup of $W$ is defined.