A singular Coxeter presentation

TL;DR

This paper introduces the singular Coxeter monoid as a category extending Coxeter groups, providing generators, relations, and a diagrammatic presentation, connecting algebraic and geometric aspects of Coxeter theory.

Contribution

It offers a new presentation of the singular Coxeter monoid, including generators, relations, and a diagrammatic approach, advancing the understanding of double cosets and their reduced expressions.

Findings

Presented a generators and relations presentation of the singular Coxeter monoid.

Described all braid relations between reduced expressions for double cosets.

Established an analogue of Matsumoto's theorem for the singular Coxeter monoid.

Abstract

We enlarge a Coxeter group into a category, with one object for each finite parabolic subgroup, encoding the combinatorics of double cosets. This category, the singular Coxeter monoid, is connected to the geometry of partial flag varieties. Our main result is a presentation of this category by generators and relations. We also provide a new description of reduced expressions for double cosets. We describe all the braid relations between such reduced expressions, and prove an analogue of Matsumoto's theorem. This gives a proper development of ideas first introduced by Geordie Williamson. In type A we also equip the singular Coxeter monoid with a diagrammatic presentation using webs.