A 2026 Seminar organised by Tom Goertzen, Tao Qin and Connor Simpson
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.
More information on the Canonical Bases in Representation Theory website.