K. N. Raghavan

Description: Speaker: K. N. Raghavan
Affiliation: The Institute of Mathematical Sciences

Date & Time: Friday, 16th February, 10:30-11:30am

Venue: Ramanujan Hall

Title: The KPRV theorem via paths
Abstract: Let V and V' be irreducible representations of a complex
semisimple Lie algebra g with highest weight vectors v and v' of weights m
and m' respectively. For w in the Weyl group, let M(m,m',w) denote the
cyclic g-submodule of V tensor V' generated by the vector v tensor wv'
(where wv' denotes a non-zero vector in V' of weight wm'). It was
conjectured by Kostant and proved by Kumar that the irreducible
representation V(m,m',w) whose highest weight is the unique dominant Weyl
conjugate of m+wm' occurs with multiplicity exactly one in the
decomposition of M(m,m',w) into irreducibles. Since M(m,m',w0) equals
V tensor V', where w0 denotes the longest element of the Weyl group, it
follows from this that V(m,m',w) occurs in the decomposition of V tensor
V'. This corollary was conjectured earlier by Parthasarathy, Ranga Rao,
and Varadarajan (PRV) and proved by Mathieu independently of Kumar.

There's a subsequent proof by Littelmann of the PRV conjecture using his
theory of Lakshmibai-Seshadri paths. I will talk about joint work with
Mrigendra Kushwaha and Sankaran Viswanath where we consider such a path
approach to Kostant's refinement of the PRV.
Location: Ramanujan Hall, Department of Mathematics
Date: Friday, February 16, 2018
Time: 10:30am-11:30am IST
Duration: 1 hour
Access: Public