Combinatorics of cone types in Coxeter groups

TL;DR

This paper explores new combinatorial properties of cone types in Coxeter groups, revealing convexity and minimal representatives, which improve computational methods for classifying elements.

Contribution

It introduces novel convexity results and minimal representatives related to cone types, enhancing understanding and computation in Coxeter groups.

Findings

Set of elements with a specific root intersection is convex in the weak order.

Existence of a unique minimal representative for these convex sets.

Improved algorithms for determining cone types of Coxeter group elements.

Abstract

In this article, we establish some new combinatorial properties of cone types in Coxeter groups. Firstly, we show that for any element $x$ in a Coxeter group $W$ and root $\beta$ in its inversion set $\Phi(x)$, the set of elements $y \in W$ satisfying $\Phi(x) \cap \Phi(y) = \{ \beta \}$ is convex in the weak order and admits a unique minimal representative. This is strongly connected to determining the cone type of elements of $W$ and leads to efficient computational methods to determine whether arbitrary elements of $W$ have the same cone type.