Geometry of certain finite Coxeter group actions

TL;DR

This paper studies the geometric and combinatorial structure of finite Coxeter group actions on multiple copies of their reflection representation, providing new insights into stratification, root subsystems, and character formulas.

Contribution

It determines a fundamental domain for the Coxeter group action on $V^{igoplus n}$ and uses this to analyze stratification, root subsystems, and characters, offering new classification and geometric results.

Findings

Established a fundamental domain for the group action.

Provided a stratification of $V^{igoplus n}$ respecting the group action.

Derived a character formula for the Coxeter group.

Abstract

We determine a fundamental domain for the diagonal action of a finite Coxeter group $W$ on $V^{\oplus n}$, where $V$ is the reflection representation. This is used to give a stratification of $V^{\oplus n}$, which is respected by the group action, and we study the geometry, topology and combinatorics of this stratification. These ideas are used to obtain results on the classification of root subsystems up to conjugacy, as well as a character formula for $W$.