Twisted Weak Orders of Coxeter Groups

TL;DR

This paper introduces the concept of twisted weak orders in Coxeter groups, linking their lattice properties to infinite reduced words and exploring conditions under which these orders form non-complete meet semilattices.

Contribution

It initiates the study of twisted weak orders, establishing their relationship with infinite reduced words and characterizing their lattice properties in affine Weyl groups.

Findings

Twisted weak orders are non-complete meet semilattices when associated with infinite reduced words.

The converse holds for affine Weyl groups, linking lattice properties to the structure of infinite reduced words.

The study connects twisted Bruhat orders with the combinatorial structure of Coxeter groups.

Abstract

In this paper, we initiate the study of the twisted weak order associated to a twisted Bruhat order for a Coxeter group and explore the relationship between the lattice property of such order and the infinite reduced words. We show that for a 2 closure biclosed set $B$ in $\Phi^+$, the $B$-twisted weak order is a non-complete meet semilattice if $B$ is the inversion set of an infinite reduced word and that the converse also holds in the case of affine Weyl groups.