Algebra Seminar
Speaker: Prof. Sankaran Viswanath (IMSc, Chennai)
Host: Krishnan Sivasubramanian
Title: A tale of two representations
Time, day and date: 11:00:00 AM, Monday, May 26
Venue: Ramanujan Hall
Abstract: We describe two well-known families of representations of the Lie algebra sl_n of traceless square matrices of size 'n' - finite-dimensional irreducible representations 'V' and tensor products of fundamental representations 'E'. Both are indexed by partitions and admit combinatorial models in terms of variants of Young tableaux. The latter module also admits an action of the infinite dimensional Lie algebra of traceless matrices whose entries are polynomials in a single variable. This ("affine") point of view gives yet another combinatorial model for 'E', in terms of Gelfand-Tsetlin patterns with a partition overlay. In this talk, we show how the finite and affine combinatorial models of 'E' are in fact related by "functorial" bijections, which respect natural projection and branching maps. We will also visualise this bijection in terms of ensembles of coloured lattice paths on a square lattice, and provide a "Proof without words" of sorts. This is based on joint work with Aritra Bhattacharya and TV Ratheesh (arXiv:2412.00116).