Subexpressions and the Bruhat order for double cosets

TL;DR

The paper extends the Bruhat order description from elements to double cosets by proving that $p \le q$ is equivalent to $p$ being the terminus of a path subordinate to a reduced expression for $q$, and establishes compatibility with concatenation via the $*$-product. It then shows that the singular Hecke 2-category admits a natural filtration by lower-term ideals compatible with the monoidal structure, defined through singular Bott-Samelson bimodules and factorization. The authors develop three intrinsic criteria for accessing these lower terms—support filtrations, factorization filtrations, and the one-tensor test—and prove their equivalence and functoriality, both in the ordinary and singular settings. Together, these results provide a robust combinatorial and categorical framework for understanding morphisms between singular Soergel bimodules and set the stage for constructing a basis of morphisms in the singular Hecke category.

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.

Theorems & Definitions (88)

• Example 1.1
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Lemma 2.7