# MSRI Summer Graduate School "Soergel Bimodules"

### Overview

Welcome to the supplementary website of the MSRI Summer Graduate School “Soergel Bimodules” organized by Ben Elias, Shotaro Makisumi, and Geordie Williamson. On the MSRI page you can find some basic information including a schedule of the lectures. Below is a more detailed description of the program.

## Exercises

# | Topic |
---|---|

1A, 1A+ | Coxeter groups |

1B, 1B+ | KL basis |

2A, 2A+ | Demazure operator, Frobenius extensions |

2B, 2B+ | Diagrammatics |

3A, 3A+ | Diagrammatics for Soergel bimodules |

3B, 3B+ | Light leaves |

4A, 4A+ | Intersection form |

4B, 4B+ | Krull–Schmidt |

5A, 5A+ | Lefschetz linear algebra |

5B, 5B+ | Category |

6A, 6A+ | Hodge theory |

6B, 6B+ | Rouquier complexes |

7A, 7A+ | Cells |

7B | Full twist |

## Book project

Below is the list of teams for our “proceedings” project. Simply download the **LaTeX template** and follow the instructions. Your chapter will become part of a book (with us as editors and you as credited authors) and we plan to publish this in the RSME Springer Series.

You should try to **finish your article by July 15** and send chapter.tex to ulrich.thiel@sydney.edu.au. We’ll then try to put everything together and you’ll get to see the final version.

*If you still would like to join, you should pick a topic marked with red as here the team is still a singleton.*

# | Topic | Team |
---|---|---|

1 | How to think about Coxeter groups I | I.Jang, F. Panelli, Y. Zhao |

2 | How to think about Coxeter groups II | G. Williamson |

3 | Hecke algebras and Kazhdan-Lusztig polynomials | J. Gibson, A. Kerschel |

4 | Category and the Kazhdan-Lusztig conjectures | L. Pasi, J. Wen |

5 | Soergel bimodules | S. Rogers, Z. Xiang |

6 | How to draw monoidal categories | A. Cepek, A. Stephens |

7 | The classical approach to Soergel bimodules | C. Bahran, E. Kowalenko |

8 |
The dihedral cathedral | N. Sandoval Gonzales |

9 |
Generators and relations, the light leaves basis | A. Sistko |

10 | How to draw Soergel bimodules | S. Shelley-Abrahamson, S. Venkatesh |

11 | Soergel’s categorification theorem | U. Thiel |

12 | Hodge theory and Lefschetz linear algebra | E. Bodish, J. Hathaway |

13 | Lightning introduction to category | A. Romanova, S. Taylor |

14 | Lightning introduction to intersection cohomology, hypercohomology and Soergel bimodules | H. Huang, K. Timochenko |

15 | Soergel’s and the Kazhdan-Lusztig conjecture | D. Matveevskiy, B. Tsvelikhovsky |

16 | The Hodge theory of Soergel bimodules | L. Taylor, M.-T. Trinh |

17 |
Rouquier complexes and homological algebra | C. Cain |

18 | Proof of hard Lefschetz | G. Dhillon, O. Kivinen |

19 |
Knot polynomials, Schur-Weyl duality, and link homology | J. Flake |

20 | Representations of the Hecke algebra and cells | S. Makisumi |

21 | Categorical diagonalization of the full twist | A. Chandler, N. Karnick, D. Vagner |

22 | Diagrammatics for Singular Soergel bimodules | B. Elias |

23 | Koszul duality I | N. Arbesfeld, V. Makam |

24 |
Koszul duality II | B. Morrissey |

25 | The situation in characteristic | G. Brown, C. Chung, C. Leonard |

26 | TBA | |

27 | TBA |

### Diagrams

Ben compiled an example file for typesetting diagrams using labelpin as explained in his lecture. You can download it here.

## Program

Each lecture is 45 minutes. There will be breaks in between and exercise sessions each day at noon and in the afternoon. The exact times of all this are listed on the MSRI page.

### Monday

**1. How to think about Coxeter groups I**

What is a Coxeter group? The ABCD of classical Weyl groups. The length function, the exchange condition.

**2. How to think about Coxeter groups II**

The Coxeter complex. Examples (dihedral groups, finite groups of Coxeter rank 3, affine groups, hyperbolic groups). The length function and the exchange condition via the Coxeter complex.

**3. Hecke algebras and Kazhdan-Lusztig polynomials**

The Coxeter complex. The Hecke algebra of a Coxeter group. The presentation using standard generators. The standard basis. The Kazhdan-Lusztig basis and polynomials. The Kazhdan-Lusztig presentation. Products of Kazhdan-Lusztig generators and the defect formula.

### Tuesday

**4. Category and the Kazhdan-Lusztig conjectures**

The happenings of 1979. The miracle of KL polynomials. Arbitrary Coxeter groups. The miracle of the localisation proof. Soergel’s dream of an algebraic explanation…the deepening mystery of positivity.

**5. Soergel bimodules**

Invariant theory for finite reflection groups. Bimodules and monoidal categories. The category of Soergel bimodules. Singular Soergel bimodules. First examples.

**6. How to draw monoidal categories**

Higher algebra. Drawing adjunctions, cyclicity etc. Example: 2-groupoids. The Coxeter groupoid. The generalized Zamolodchikov relations.

### Wednesday

**7. The classical approach to Soergel bimodules**

Standard bimodules. Support filtrations. Soergel’s hom formula. Statement of Soergel’s categorification theorem. Localization. Discussion.

**8. The dihedral cathedral**

Starting to draw Soergel bimodules. Soergel bimodules in rank 2. Jones-Wenzl projectors, connections to the Temperley-Lieb algebra and quantum groups. Categorification of the Kazhdan-Lusztig presentation.

**9. Generators and relations, the light leaves basis**

Generators and relations in general. Light leaves morphisms as a categorification of the defect formula. Double leaves give a basis for morphisms.

### Thursday

**10. How to draw Soergel bimodules**

Drawing Bott-Samelson bimodules, Soergel bimodules. Intersection forms. Discussion.

**11. Soergel’s categorification theorem**

The cellular structure. A discussion of idempotent lifting. Generators and relations proof of Soergel’s categorification theorem. Examples of intersection forms and idempotents.

**12. Hodge theory and Lefschetz linear algebra**

Review of the (real) Hodge theory of smooth projective algebraic varieties. A discussion of the weak and hard Lefschetz theorems. Lefschetz operators, Lefschetz forms and the Hodge-Riemann bilinear relations. Tricks establishing the Lefschetz package. The weak-Lefschetz substitute.

### Friday

**13. Lightning introduction to category **

Category , Verma modules, translation functors, block decomposition.

**14. Lightning introduction to intersection cohomology, hypercohomology and Soergel bimodules**

Varieties stratified by affine spaces and the constructible derived category. How to compute stalks of a proper push-forward. Poincaré duality. Stalks definition of an IC sheaf. The connection to Kazhdan-Lusztig polynomials on the flag variety. Global sections and Soergel bimodules.

**15. Soergel’s and the Kazhdan-Lusztig conjecture**

Statement of the Kazhdan-Lusztig conjecture. Soergel’s functor . Soergel’s conjecture implies the Kazhdan-Lusztig conjecture.

### Monday

**16. The Hodge theory of Soergel bimodules**

Statement of the results and outline of the methods. The embedding theorem, the limit argument. The absence of the weak Lefschetz theorem.

**17. Rouquier complexes and homological algebra**

The homotopy category of Soergel bimodules. Minimal complexes. Rouquier complexes. Examples.

**18. Proof of hard Lefschetz**

The perverse filtration on Soergel bimodules. The diagonal miracle. Factoring the Lefschetz operator. Hard Lefschetz.

### Tuesday

Independence day!

### Wednesday

**19. Knot polynomials, Schur-Weyl duality, and link homology**

Polynomials attached to braid closures coming from representation theory. Schur-Weyl duality, and why these polynomials can be computed using the Hecke algebra. How to categorify this theory.

**20. Representations of the Hecke algebra and cells**

Young-Jucys-Murphy elements, full twist braids, and their eigenvalues. Cellular quotients of the Hecke algebra, and the relationship between link homologies. Starting to categorify these ideas.

**21. Categorical diagonalization of the full twist**

Lagrange interpolation and the construction of projections to eigenspaces. Categorical diagonalization. Applying this to the full twist Rouquier complex. Applications to link homology.

### Thursday

**22. Diagrammatics for Singular Soergel bimodules**

Further discussion of Frobenius extensions, bases and dual bases. Frobenius (hyper)cubes and their relations. The Schur category and its categorification via singular Soergel bimodules. Further relations.

**23. Koszul duality I**

Koszul duality for polynomial rings. Historical motivation, inversion formula for Kazhdan-Lusztig polynomials, two approaches to Kazhdan-Lusztig conjecture.

**24. Koszul duality II**

Bezrukavnikov-Yun and monoidal Koszul duality. The right equivariant, left monodromic and free monodromic categories.

### Friday

**25. The situation in characteristic **

Lusztig’s conjecture. Intersection forms. The -canonical basis. Examples.

**26. TBA (see below)**

**27. TBA (see below)**

### Other lectures

Determined by popular demand from the following list.

**1. Categorifications of braid groups**

Categorifying the braid group. Example of Rouquier complexes. Generators and relations for strict braid group actions. Deligne’s theorem and the EW version.

**2. Hecke algebras with unequal parameters and folding**

Definition of Hecke algebras with unequal parameters. Equivariant K-theory. Categorification of unequal parameters in the quasi-split case.

**3. Algebraic quantum geometric Satake**

A discussion of ridiculous titles. Algebraizing the geometric Satake equivalence. Quantizing it in type A using Ben’s favorite Cartan matrix.

**4. Torsion explosion**

A very complicated way of producing large numbers.

**5. Relative hard Lefschetz for Soergel bimodules**

Statement of relative hard Lefschetz. Combinatorial consequences. Idea of proof.

**6. Lusztig’s -ring and its categorification**

What is the -ring, and why does one care. How to categorify it. Rigidity.

**7. Local Hodge theory and the Jantzen conjectures**

The Jantzen and Andersen filtrations. What are the Jantzen conjectures and why is it amazing? Local Hodge theory of Soergel bimodules. Examples.

**8. Manin-Schechtmann theory and idempotents in type **

Graded orders on the reduced expression graph. Higher Bruhat orders. Path morphisms and idempotents.

**9. Lightning introduction to intersection cohomology, hypercohomology**

This was meant to be part of lecture 14