Diagrammatic algebra in representation theory

Instructor: Alexis Langlois-Rémillard

Examination: Exams will be oral on topics of the course; they will take place on the last week of July (28-31 July). Later in the semester you will be able to book a 45min spot to take it.

A preliminary plan is available here.

Schedule

The 90 min lectures will take place Mondays 8:00 to 10:00 in Zeichensaal, Wegelerstr. 10. (The precise distribution of the 90 min in the alloted time will be decided upon during the first course.)

Week-by-week schedule

More information to come.

07.04.2025 Lecture 1: Administration, book distribution, some preliminiaries and our first diagrammatic groups. (Bowman, Chapters 1-2)

14.04.2025 Lecture 2: Temperley-Lieb algebras

21.04.2025 No lecture due to Ostermontag

28.04.2025 Lecture 3: (Jones normal form) and Diagrammatic algebras coming from mathematical physics

05.05.2025 Lecture 4: Cellular algebra

12.05.2025 Lecture 5: Grading, weight and idempotents

19.05.2025 Lecture 6: A detour on monoidal categories

26.05.2025 Lecture 7: [Dates to change] Combinatorial Kazhdan-Lusztig theory I

02.06.2025 Lecture 8: Combinatorial Kazhdan-Lusztig theory II

09.06.2025 No lecture due to Pfingstmontag

16.06.2025 Lecture 9: Counterexample to Lusztig conjecture I

23.06.2025 Lecture 10: Counterexample to Lusztig conjecture II

30.06.2025 Lecture 11: [Date to change] Counterexample to Lusztig conjecture III

07.07.2025 Lecture 12: [Date to change] A KLR algebra I

14.06.2025 Lecture 13: A KLR algebras II

Course notes

14.04.2025 In this lecture, we followed Ridout and Saint-Aubin. Standard modules, induction and the structure of the Temperley-Lieb algebra. Adv. Theo. Math. Phys. 18.5 pp 957-1041 (2014) https://dx.doi.org/10.4310/ATMP.2014.v18.n5.a1.

Notes of the lectures will appear here. We will be roughly following Chris Bowman's book «Diagrammatic algebra» (to appear). In general, I will only provide notes when I depart from the book. A (virtual) copy of the book will be shared with students at the first lecture (thanks Chris!) and we will take bits and pieces from other sources. You can email me for the pdf of the course.

Problem sheets

07.04.2025 Problem sheet 1

14.04.2025 Problem sheet 2 (updated 17.04.2025 to correct a small typo)

Each week, a few problems will be given out. I do not expect you to do all of them, but I will assume you read them. Exercises are either from the book or from different sources. I will be more than happy to discuss solutions either at the office, before or after the course, or via email.