Subexpressions and the Bruhat order for double cosets

TL;DR

This paper extends the understanding of the Bruhat order from Coxeter groups to double cosets, establishing fundamental properties and compatibility with algebraic structures in the singular Hecke 2-category.

Contribution

It introduces an analogous subexpression description for the Bruhat order on double cosets and proves its compatibility with monoidal structures.

Findings

Bruhat order on double cosets described via subexpressions

Compatibility established between Bruhat order and monoidal structure

Fundamental properties of the ideal of lower terms proved

Abstract

The Bruhat order on a Coxeter group is often described by examining subexpressions of a reduced expression. We prove that an analogous description applies to the Bruhat order on double cosets. This establishes the compatibility of the Bruhat order on double cosets with concatenation, leading to compatibility between the monoidal structure and the ideal of lower terms in the singular Hecke 2-category. We also prove other fundamental properties of this ideal of lower terms.