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Soergel Calculus with patches

Leonardo Maltoni

Abstract

We adapt the diagrammatic presentation of the Hecke category to the dg category formed by the standard representatives for the Rouquier complexes. We use this description to recover basic results about these complexes, namely the categorification of the relations of the braid group and the Rouquier formula.

Soergel Calculus with patches

Abstract

We adapt the diagrammatic presentation of the Hecke category to the dg category formed by the standard representatives for the Rouquier complexes. We use this description to recover basic results about these complexes, namely the categorification of the relations of the braid group and the Rouquier formula.
Paper Structure (17 sections, 10 theorems, 44 equations)

This paper contains 17 sections, 10 theorems, 44 equations.

Table of Contents

  1. Review of the diagrammatic Hecke category
  2. Coxeter systems, braid groups and Hecke algebras
  3. Realizations of Coxeter systems
  4. The diagrammatic Hecke category
  5. Localization
  6. Light leaves and double leaves
  7. Diagrammatics for dg Rouquier complexes
  8. Rouquier complexes
  9. Basic properties of Rouquier complexes
  10. Notation for graded objects and complexes
  11. A dg monoidal category of Rouquier complexes
  12. Diagrammatics
  13. DG double leaves
  14. Inverse relation
  15. Braid relation
  16. ...and 2 more sections

Key Result

Theorem 1.3

If $\Bbbk$ is a complete local ring, then $\mathscr{H}$ is Krull-Schmidt and there is a unique isomorphism of $\mathbb{Z}[v,v^{-1}]$-algebras, called character, sending the class of $B_s$ to $b_s$.

Theorems & Definitions (40)

  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition
  • Example 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Proposition 2.1
  • ...and 30 more