On convex hulls of orbits of Coxeter groups and Weyl groups

TL;DR

This paper studies the convex hulls of orbits of Coxeter and Weyl groups, showing they are determined by reflections and extending results to various root systems with applications in representation theory.

Contribution

It proves that the convex hull of a Coxeter group orbit is determined by the orbit's image under reflections, extending to Weyl groups of various root systems.

Findings

Convex hulls of orbits are determined by reflections.

Results apply to locally finite and affine root systems.

Applications include minimizing linear functionals on orbits.

Abstract

The notion of a linear Coxeter system introduced by Vinberg generalizes the geometric representation of a Coxeter group. Our main theorem asserts that if $v$ is an element of the Tits cone of a linear Coxeter system and $\cW$ is the corresponding Coxeter group, then $\cW v \subeq v - C_v,$ where $C_v$ is the convex cone generated by the coroots $\check \alpha$, for which $\alpha(v) > 0$. This implies that the convex hull of $\cW v$ is completely determined by the image of $v$ under the reflections in $\cW$. We also apply an analogous result for convex hulls of $\cW$-orbits in the dual space, although this action need not correspond to a linear Coxeter system. Motivated by the applications in representation theory, we further extend these results to Weyl group orbits of locally finite and locally affine root systems. In the locally affine case, we also derive some applications on minimizing linear functionals on Weyl group orbits.