Combinatorics of cone types in Coxeter groups

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 ${\hat{I}}^2$ in its inversion set $\hat{I}¦(x)$, the set of elements $y\in W$ satisfying $\hat{I}¦(x)\cap \hat{I}¦(y)=\{{\hat{I}}^2\}$ 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.

We were made aware that Theorem 1 in the previous version is Theorem 8.1 in Reading and Speyer, "Sortable elements in infinite Coxeter groups, Transactions of the American Mathematical Society (2011), so the paper has been reorganised and exposition has been updated. Our alternative proof of Theorem 8.1 of Reading and Speyer has been repositioned to the last complimentary section