Canonical Bases in Representation Theory
at the University of Sydney, SMRI Seminar Room
Seminar meeting: Wednesdays 10-12
Exercise sessions: Wednesdays 13-14
First meeting: March 4 2026
at the University of Sydney, SMRI Seminar Room
Seminar meeting: Wednesdays 10-12
Exercise sessions: Wednesdays 13-14
First meeting: March 4 2026
In this seminar, we aim to study the answers of the following motivating questions regarding irreducible representations of semisimple Lie algebras and related structures:
How can we compute their characters?
How can we compute tensor product decompositions?
What are "canonical bases" for these modules?
A powerful tool introduced in the early 1990s, known as the canonical basis (Lusztig) or crystal basis (Kashiwara), provides a model to answer these questions.
We will start by learning about prototypical constructions that serve as motivation, before proceeding to the construction and properties of these modern bases: first studying Lusztig's approach, and then Kashiwara's approach.
This seminar will focus on the main ingredients and recipes used to motivate, construct and describe these bases: Kazhdan-Lusztig bases, Gelfand-Tsetlin bases, Lusztig's algebraic construction, Lusztig's geometric/topological construction, and Kashiwara's crystal/global bases.
You can let us (the organizers) know if you would like to be added to the mailing list and/or would like to give a talk (see bottom of the website for contact).
The seminar will meet on Wednesdays from 10-12 in the SMRI seminar room. There will be a weekly exercise session on Wednesdays 1-2pm, again in the SMRI seminar room.
The following is a tentative schedule. The rough plan: first part of the course we will motivate and understand prototypical constructions, next we learn about Lusztig's approach and finally we go to Kashiwara's approach. The focus will always be on the big picture and motivating examples, especially in low ranks. Below you can find a tentative schedule
March 4: Introduction of the motivation and framework + Kazhdan-Lusztig Bases
March 11: Gelfand-Tsetlin Bases
March 18: Introduction to Quantum Groups
March 25: Lusztig's algebraic construction (Part I, Tools and Methods)
April 1: Lusztig's algebraic construction (Part II, Tools, Methods and Canonical Basis)
April 15: Properties of the Canonical Basis
April 22: Geometry behind Canonical Bases
April 29: TBD (The remaining lectures will discuss the crystal basis and further topics)
May 6: TBD
May 13: TBD
May 20: TBD
May 27: TBD
June 3: TBD
...: TBD
Chari, V., & Pressley, A. (1994). A Guide to Quantum Groups. Cambridge University Press.
Hong, J., & Kang, S.-J. (2002). Introduction to Quantum Groups and Crystal Bases (Graduate Studies in Mathematics, Vol. 42). American Mathematical Society.
Lusztig, G. (1993). Introduction to Quantum Groups (Progress in Mathematics, Vol. 110). Birkhäuser.
Curtis, C. W. (2012). New directions in representation theory. Milan Journal of Mathematics, 80, 151–167.
https://doi.org/10.1007/s00032-012-0177-8
Kamnitzer, J. (2022). Perfect bases in representation theory. Proceedings of the International Congress of Mathematicians (ICM 2022).
Video | Slides (PDF)
Kashiwara, M. (1991). On crystal bases of the q-analogue of universal enveloping algebras. Duke Mathematical Journal, 63(2), 465–516.
Lusztig, G. (1990). Canonical bases arising from quantized enveloping algebras. Journal of the American Mathematical Society, 3(2), 447–498.
Lusztig, G. (1991). Quivers, perverse sheaves, and quantized enveloping algebras. Journal of the American Mathematical Society, 4(2), 365–421.
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