Reciprocity and rationality for the greedy normal form of a Coxeter group

TL;DR

This paper proves the rationality of the characteristic series for the greedy normal form of Coxeter groups and establishes a reciprocity formula for right-angled groups with Eulerian nerve, extending known results.

Contribution

It introduces a new rationality result for the greedy normal form series and a reciprocity formula for specific Coxeter groups, unifying and extending prior findings.

Findings

Characteristic series for the greedy normal form is always rational.

Reciprocity formula established for right-angled Coxeter groups with Eulerian nerve.

Extends known rationality and reciprocity results for Coxeter group growth series.

Abstract

We show that the characteristic series for the greedy normal form of a Coxeter group is always a rational series, and prove a reciprocity formula for this series when the group is right-angled and the nerve is Eulerian. As corollaries we obtain many of the known rationality and reciprocity results for the growth series of Coxeter groups as well as some new ones.