Coxeter groups, imaginary cones and dominance

TL;DR

This paper explores the relationship between dominance and $ $-height in Coxeter groups, revealing that $ $-height equals the number of positive roots dominated by a reflection, and applies this to study imaginary cones.

Contribution

It establishes the equivalence between $ $-height and the count of dominated positive roots, linking two concepts in Coxeter group theory and applying this to imaginary cone analysis.

Findings

$ $-height of a reflection equals the number of dominated positive roots

Finite $ $-height sets are finite for finite rank Coxeter groups

Applications to the structure of imaginary cones in Coxeter groups

Abstract

Brink and Howlett have introduced a partial ordering, called dominance, on the positive roots in the Tits realization of Coxeter groups (Math. Ann. 296 (1993), 179--190). Recently a concept called $\infty$-height is introduced to each reflection in an arbitrary Coxeter group $W$ (Edgar, Dominance and regularity in Coxeter groups, PhD thesis, 2009). It is known (Dyer, unpublished) that for all $W$ of finite rank, and for each non-negative $n$, the set of reflections of $\infty$-height equal to $n$ is finite. However, it is not clear that the concepts of $\infty$-height and dominance are related. Here we show that the $\infty$-height of an arbitrary reflection is equal to the number of positive roots strictly dominated by the positive root corresponding to that reflection. We also give applications of dominance to the study of imaginary cones of Coxeter groups.