Bruhat Preclosure

TL;DR

This paper investigates the properties of the Bruhat preclosure in Coxeter groups, proving Dyer's conjecture in type A and establishing the infinite Bruhat closure as a key concept for understanding inversion sets and joins.

Contribution

It introduces the Bruhat preclosure and infinite Bruhat closure, providing new insights and proofs related to Dyer's conjecture in Coxeter groups, especially in type A.

Findings

Bruhat preclosure is a preclosure, becoming a closure in inversion sets.

Infinite Bruhat closure is obtained by iterating the preclosure.

Dyer's conjecture holds in type A using the Bruhat preclosure.

Abstract

In 2011, Dyer published a series of conjectures on the weak order of Coxeter groups. One of these conjectures stated that the inversion set of the join of two elements in a Coxeter group is equal to some "closure" of the union of their inversion sets. In this paper we show that this "closure" is in fact a preclosure, which we call the Bruhat preclosure, but is a closure whenever our underlying set is an inversion set. By performing the Bruhat preclosure an infinite number of times we obtain a closure which we call the infinite Bruhat closure. We show in a uniform way that Dyer's conjecture is true when using the infinite Bruhat closure (instead of Bruhat preclosure) if the join exists between two elements. Finally, we end by showing in type A, the Bruhat preclosure is a closure thus giving a (second) proof that Dyer's conjecture is true in type A.