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Singular Light Leaves

Ben Elias, Hankyung Ko, Nicolas Libedinsky, Leonardo Patimo

Abstract

For any Coxeter system we introduce the concept of singular light leaves, answering a question of Williamson raised in 2008. They provide a combinatorial basis for Hom spaces between singular Soergel bimodules.

Singular Light Leaves

Abstract

For any Coxeter system we introduce the concept of singular light leaves, answering a question of Williamson raised in 2008. They provide a combinatorial basis for Hom spaces between singular Soergel bimodules.
Paper Structure (53 sections, 50 theorems, 250 equations, 1 algorithm)

This paper contains 53 sections, 50 theorems, 250 equations, 1 algorithm.

Table of Contents

  1. Background material and Grassmannian reduction
  2. Double cosets, expressions, and Grassmannian pairs
  3. Expressions for double cosets
  4. Redundancy
  5. Reduction to Grassmannian pairs
  6. Proof of reduction to Grassmannian pairs
  7. The core of a double coset
  8. The Hecke algebroid
  9. Basics and bases
  10. Generators of the Hecke algebroid
  11. The singular Deodhar formula
  12. Singular Soergel bimodules
  13. Assumptions and notation
  14. Definitions
  15. Classification of singular Soergel bimodules
  16. ...and 38 more sections

Key Result

Theorem 1.6

Let $p \subset q$ be a coset pair for $(I,J,s)$. Let $\mathop{\mathrm{RQ}}\nolimits = \mathop{\mathrm{RR}}\nolimits(q)$, and let $n$ be the $(\mathop{\mathrm{RQ}}\nolimits,Js)$-coset whose underlying set is $W_{Js}$. Then there exists an $(I,\mathop{\mathrm{RQ}}\nolimits)$-coset $z$ and a $(\mathop{ More precisely, $m, n$ and $z$ are determined by

Theorems & Definitions (178)

  • Definition 1.3
  • Definition 1.4
  • Example 1.5
  • Theorem 1.6
  • Example 1.8
  • Proposition 1.9
  • proof : Proof of Theorem \ref{['thmB']} and Proposition \ref{['thmBconti']}
  • Example 1.10
  • Remark 1.11
  • Remark 2.1
  • ...and 168 more