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Demazure operators for double cosets

Ben Elias, Hankyung Ko, Nicolas Libedinsky, Leonardo Patimo

TL;DR

The work develops a rigorous algebraic framework for Demazure operators tied to double cosets in a Coxeter group, culminating in the nilCoxeter algebroid $\mathcal{D}$ whose morphisms are $\{\partial_p\}$ indexed by double cosets. It proves a Matsumoto-type reduction theorem for singular expressions and provides a concrete presentation of $\mathcal{D}$ by generators and relations, mirroring classical nilCoxeter theory. A generalized Demazure surjectivity result yields graded Frobenius extensions $R^I\subset R$ and $R^J\subset R^I$, with explicit almost dual bases and dual-basis-in-image phenomena that will underpin a diagrammatic calculus for singular Bott-Samelson bimodules. The results connect to cohomology/K-theory interpretations and set the stage for applications to singular Soergel bimodules and their Bott-Samelson resolutions, offering a robust algebraic toolkit for categorified representation theory.

Abstract

For any Coxeter system, and any double coset for two standard parabolic subgroups, we introduce a Demazure operator. These operators form a basis for morphism spaces in a category we call the nilCoxeter category, and we also present this category by generators and relations. We prove a generalization to this context of Demazure's celebrated theorem on Frobenius extensions. This generalized theorem serves as a criterion for ensuring the proper behavior of singular Soergel bimodules.

Demazure operators for double cosets

TL;DR

The work develops a rigorous algebraic framework for Demazure operators tied to double cosets in a Coxeter group, culminating in the nilCoxeter algebroid whose morphisms are indexed by double cosets. It proves a Matsumoto-type reduction theorem for singular expressions and provides a concrete presentation of by generators and relations, mirroring classical nilCoxeter theory. A generalized Demazure surjectivity result yields graded Frobenius extensions and , with explicit almost dual bases and dual-basis-in-image phenomena that will underpin a diagrammatic calculus for singular Bott-Samelson bimodules. The results connect to cohomology/K-theory interpretations and set the stage for applications to singular Soergel bimodules and their Bott-Samelson resolutions, offering a robust algebraic toolkit for categorified representation theory.

Abstract

For any Coxeter system, and any double coset for two standard parabolic subgroups, we introduce a Demazure operator. These operators form a basis for morphism spaces in a category we call the nilCoxeter category, and we also present this category by generators and relations. We prove a generalization to this context of Demazure's celebrated theorem on Frobenius extensions. This generalized theorem serves as a criterion for ensuring the proper behavior of singular Soergel bimodules.
Paper Structure (21 sections, 29 theorems, 75 equations)

This paper contains 21 sections, 29 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Singular reduced expressions and the Matsumoto theorem
  3. Some notation
  4. Expressions and reduced expressions
  5. Concatenation
  6. Redundancy and the core
  7. Addable and removable elements
  8. Braid relations and Matsumoto's theorem
  9. Non-reduced expressions
  10. Demazure operators and double cosets
  11. Realizations
  12. Basics of Demazure operators
  13. NilCoxeter algebra as associated graded
  14. The nilCoxeter algebroid
  15. Aside on cohomology and K-theory
  16. ...and 6 more sections

Key Result

Proposition 2.7

Let $I_{\bullet} \leftrightharpoons p$ and $K_{\bullet} \leftrightharpoons q$ be composable reduced expressions, with $J := K_0 = I_d$. Then $I_{\bullet} \circ K_{\bullet}$ is reduced if and only if MoreoverThis last statement is only stated implicitly in EKo, and follows immediately from unraveling the equality $p * q = r$., $I_{\bullet} \circ K_{\bullet} \leftrightharpoons r$ where

Theorems & Definitions (92)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 82 more