The Euler characteristic of a Hecke algebra

TL;DR

This paper proves that the Euler characteristic of a Hecke algebra associated with a Coxeter group equals the inverse of its Poincaré series, linking algebraic and combinatorial invariants.

Contribution

It establishes a precise relationship between the Euler characteristic of Hecke algebras and the Poincaré series of Coxeter groups, a novel connection in algebraic combinatorics.

Findings

Euler characteristic equals inverse Poincaré series

Links algebraic invariants with combinatorial properties

Provides a new formula for Hecke algebra invariants

Abstract

It is shown that the Euler characteristic $\chi_{(\mathcal{H},\mathcal{B},\epsilon_q)}$ of a $\mathbb{Z}[[q]]$-Hecke algebra $\mathcal{H}$ associated with a finitely generated Coxeter group $(W,S)$ coincides with $p_{(W,S)}(q)^{-1}$, where $p_{(W,S)}(t)$ is the Poincar\'e series of $(W,S)$.