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Categorical diagonalization of full twists

Ben Elias, Matthew Hogancamp

Abstract

We conjecture that the complex of Soergel bimodules associated with the full twist braid is categorically diagonalizable, for any finite Coxeter group. This utilizes the theory of categorical diagonalization introduced earlier by the authors. We prove our conjecture in type $A$, and as a result we obtain a categorification of the Young idempotents.

Categorical diagonalization of full twists

Abstract

We conjecture that the complex of Soergel bimodules associated with the full twist braid is categorically diagonalizable, for any finite Coxeter group. This utilizes the theory of categorical diagonalization introduced earlier by the authors. We prove our conjecture in type , and as a result we obtain a categorification of the Young idempotents.

Paper Structure

This paper contains 81 sections, 135 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Representation theory of symmetric groups
  3. Lifting to the Hecke algebra
  4. The categorification
  5. Categorical linear algebra
  6. The main theorem
  7. Representations and cells
  8. Cell filtrations and diagonalization
  9. Homological triangularity
  10. Bounding the action of the full twist
  11. Constructing $\lambda$-equivalences
  12. Ensuring relative diagonalization
  13. Tightness
  14. Other Coxeter groups
  15. Relation to other work
  16. ...and 66 more sections

Key Result

Theorem 1.2

(See ElHog17a for a precise version) Suppose that $F\in{\mathcal{K}}^b({\mathcal{A}})$ is invertible. Let $I$ be a finite set and suppose there is an $I$-indexed family of invertible scalar objects $\Sigma_\lambda$ with distinct homological shifts. Suppose there are chain maps $\alpha_\lambda\colon Then under additional technical hypothesesThese hypotheses imply that the cones $\operatorname{Cone

Theorems & Definitions (343)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 1.10
  • ...and 333 more