Braid groups of normalizers of reflection subgroups

TL;DR

This paper investigates the structure of braid groups associated with normalizers of reflection subgroups in finite complex reflection groups, establishing split extensions and bases for related Hecke algebras, especially in Coxeter groups.

Contribution

It proves the extension splits for Coxeter groups and constructs a standard basis for the associated Hecke algebra, extending understanding of these algebraic structures.

Findings

Extension splits in Coxeter groups

Standard basis for the Hecke algebra $ ilde{H}_0$

Examples of split and non-split cases in non-Coxeter groups

Abstract

Let $W_0$ be a reflection subgroup of a finite complex reflection group $W$, and let $B_0$ and $B$ be their respective braid groups. In order to construct a Hecke algebra $\widetilde{H}_0$ for the normalizer $N_W(W_0)$, one first considers a natural subquotient $\widetilde{B}_0$ of $B$ which is an extension of $N_W(W_0)/W_0$ by $B_0$. We prove that this extension is split when $W$ is a Coxeter group, and deduce a standard basis for the Hecke algebra $\widetilde{H}_0$. We also give classes of both split and non-split examples in the non-Coxeter case.