The Poincar\'e Series of Coxeter Folding Subgroups

TL;DR

This paper investigates the length distribution of folding subgroups within Coxeter groups, deriving explicit generating functions that reveal new combinatorial identities involving length statistics.

Contribution

It provides explicit formulas for the length generating functions of folding subgroups in finite and affine Coxeter groups, connecting them to q-integers and combinatorial identities.

Findings

Derived explicit length generating functions for folding subgroups

Revealed combinatorial identities involving length statistics

Connected generating functions to q-integers and polynomial identities

Abstract

Folding subgroups give a way to realize non-simply-laced Coxeter groups as subgroups of simply-laced Coxeter groups. In this paper, we study how folding subgroups of finite and affine type are distributed length-wise by calculating the length generating function of the subgroup with respect the length of the ambient group. These generating functions have surprisingly nice formulas in terms of $q$-integers and give rise to interesting combinatorial identities on polynomials involving length statistics of both the ambient group and folding subgroup.