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.

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.

Description

Ramanujan Hall, Department of Mathematics

Date

Fri, February 16, 2018

Start Time

10:30am-11:30am IST

Duration

1 hour

Priority

5-Medium

Access

Public

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Updated

Sun, February 11, 2018 12:28pm IST