Invariant convex subcones of the Tits cone of a linear Coxeter group

TL;DR

This paper studies the structure of convex subcones of the Tits cone in linear Coxeter groups, focusing on their faces and face lattices, and generalizes previous results to broader classes of Coxeter systems.

Contribution

It characterizes the faces and face lattices of Coxeter group invariant convex subcones, extending known results to more general Coxeter systems and cones.

Findings

Determined the face lattice of the Tits cone.

Analyzed the imaginary cone and its faces.

Generalized previous results for symmetric and free root bases.

Abstract

We investigate the faces and the face lattices of arbitrary Coxeter group invariant convex subcones of the Tits cone of a linear Coxeter system as introduced by E. B. Vinberg. Particular examples are given by certain Weyl group invariant convex cones which underlie the theory of normal reductive linear algebraic monoids as developed by M. S. Putcha and L. E. Renner. We determine the faces and the face lattice of the Tits cone and the imaginary cone, generalizing some of the results obtained for linear Coxeter systems with symmetric root bases by M. Dyer, and for linear Coxeter systems with free root bases by E. Looijenga, P. Slodowy, and the author.