Reflection length in the general linear and affine groups

TL;DR

This paper provides an intrinsic criterion for reduced reflection factorizations in the general linear group and introduces a formula for reflection length in the affine group, advancing understanding of their algebraic structure.

Contribution

It offers a new criterion for reflection factorizations in linear groups and a formula for reflection length in affine groups, which were previously not well-understood.

Findings

Intrinsic criterion for reduced reflection factorizations in GL(n)

Formula for reflection length in affine groups

Enhanced understanding of reflection structures in linear and affine groups

Abstract

We give an intrinsic criterion to tell whether a reflection factorization in the general linear group is reduced, and give a formula for computing reflection length in the general affine group.