The Deletion Order and Coxeter Groups

TL;DR

This paper explores the deletion order in Coxeter groups, revealing its properties, relation to Bruhat order, and applications in constructing tilings and understanding group elements.

Contribution

It introduces the deletion order as a refinement of Bruhat order and demonstrates its use in defining normal forms and analyzing Coxeter group properties.

Findings

Deletion order refines Bruhat order in Coxeter groups

Normal forms of elements coincide with lexicographic normal forms

Results connect deletion order with Artinian property and longest element

Abstract

The deletion order of a finitely generated Coxeter group W is a total order on the elements which, as is proved, is a refinement of the Bruhat order. This order is applied in [8] to construct Elnitsky tilings for any finite Coxeter group. Employing the deletion order, a corresponding normal form of an element w of W is defined which is shown to be the same as the normal form of w using right to left lexicographic ordering. Further results on the deletion order are obtained relating to the property of being Artinian and, when W is finite, its interplay with the longest element of W.