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. (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 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: 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

    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

    • 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.