Seminar in Representation Theory
Fluid dynamics in asymptotic representation theory
2025/02/25 (Tue.) 15:30~17:00
- Date : 2025/02/25 (Tue.) 15:30~17:00
- Location : Seminar Room 638, Institute of Mathematics (NTU Campus)
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Speaker : Colin McSwiggen (AS)
- Organizer : Cheng-Chiang Tsai (AS)
- Abstract : Many contemporary questions in probability and physics hinge on understanding the representation theory of Lie groups of very large rank. In this talk, we ask how the weight multiplicities of irreducible representations of compact Lie groups behave as the rank of the group grows large. To answer this question, we relate the characters of irreducible representations to the transition kernel of a certain stochastic interacting particle system, where the rank of the group corresponds to the number of particles. The large-rank asymptotics of the characters are then controlled by the hydrodynamic limit of this particle system, that is, the limiting regime in which the particles behave like a fluid. Thus we derive a formula for the large-rank limit of weight multiplicities in terms of solutions to this fluid system. We also obtain an analogous formula for the large-N limit of generalized Bessel functions arising in Dunkl theory. As time allows, I'll speculate recklessly about further applications. All of the results that I'll present are drawn from this joint paper with Jiaoyang Huang: https://arxiv.org/abs/2305.04131
