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.

Office hours: Monday 10-11 (just after the course), or on appointement. Office 4.042 of Math Zentrum. (If you want to take your chance you can come to my office Monday-Thursday 11-17 and I should be there, but send an email to be sure as I sometimes go to seminar and work elsewhere.)

Schedule

The 90 min lectures will take place Mondays 8:00 to 10:00 in Zeichensaal, Wegelerstr. 10. Precisely, the lecture take place from 8:15-9:00, health break, 9:05 to 9:50.

Week-by-week schedule

  • 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: [Was: Grading, weight and idempotents -> Lect. 6] Became Cellular algebra II
  • 19.05.2025 Lecture 6: [Was: A detour on monoidal categories -> Lect 13] Replaced by: Grading, weight and idempotents
  • 26.05.2025 Lecture 7: NO COURSE: The class is open and I encourage you to come and work together on the problem sheets!
  • 02.06.2025 Lecture 8: Combinatorial Kazhdan-Lusztig theory I and evaluation! Bring a mobile device or a computer https://www.umfragen.uni-bonn.de/lehre/Mathe/
  • 09.06.2025 No lecture due to Pfingstmontag
  • 16.06.2025 Lecture 9: CLASS CANCELLED
  • 23.06.2025 Lecture 10: Combinatorial Kazhdan-Lusztig theory II C
  • 30.06.2025 Lecture 11: Recorded 44min lecture and reading on General Kazhdan-Lusztig
  • 07.07.2025 Lecture 12: Recorded 30min lecture and reading on the Hecke category
  • 14.07.2025 Lecture 13: Soergel diagram and categorification
  • 21.07.2025 Lecture 14: A final detour on monoidal categories (at 10:00-12:00)
  • 22.07.2025 Extra lecture by Chris Bowman on the Counterexample of Lusztig Conjecture (PM, precise date and location to decide later)
  • Course notes

  • 19.05.2025:Lecture notes
  • The following lecture notes give roughly what we followed from the book and provide the extra material I extracted from other sources (Mainly Mathas' book Iwahori-Hecke algebras and Schur algebras of the symmetric group).

    Remarks

    30-06-2025 and 07-07-2025 There won't be any lecture. However, recorded snippets on the material will be available on Scibo file: https://uni-bonn.sciebo.de/s/tPBFi7A4MYL3kgr .

    23.06.2025 Chris Bowman agreed to give a closing lecture on the Counterexample to the Lusztig Conjecture! It will be a 2h lecture in the afternoon of July 22nd. (Small note, this material is outside the evaluation, so it will be pure enjoyment.)

    16.06.2025 Class cancelled

    05.05.2025 We covered a bit of cellular algebra theory. I've written up the notes in the exercise sheet with some extra details on the proofs we did not do in class.

    28.04.2025 In this lecture, I have given a bit on the physical motivation of the TL algebras and variations. It's a bit hard to suggest good reference on this. I suggest: P. Martin, Potts Models and Related Problems in Statistical Mechanics, Volume 5 of Advances in Statistical Mechanics. World Scientific,Singapore, 1991. and R. Baxter, Exactly Solved Models in Statistical Mechanics. Academic Press, London, 1982. The scanned course notes might help.

    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)
  • 28.04.2025 Problem sheet 3
  • 05.05.2025 Problem sheet 4 with extra material
  • 12.05.2025 Problem sheet 5 with extra material
  • 19.05.2025 Problem sheet 6
  • 02.06.2025 Problem sheet 7
  • 23.06.2025 Problem sheet 8
  • 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.