#### IMSc Webinar

#### A study of Kostant-Kumar modules via Littelmann paths

#### Mrigendra Singh Kushwaha

##### IMSc Chennai

*Kostant-Kumar modules are certain cyclic submodules of the tensor product*

of two irreducible integrable highest-weight modules of a symmetrizable Kac-Moody algebra.

We give, in the spirit of Littelmann, a path model for Kostant-Kumar modules in terms of

Lakshmibai-Seshadri paths.

Littelmann’s path model gives a generalized Littlewood-Richardson rule for decomposing

tensor products into irreducibles. An analogous rule for Kostant-Kumar modules was given

by Joseph under the hypothesis that the Kac-Moody algebra is symmetric. We extend this

to finite type Lie algebras and use this rule to study Parthasarathy-Ranga Rao-Varadarajan

(PRV) components and generalized PRV components in Kostant-Kumar modules.

At the end, we discuss Kostant-Kumar modules for the finite dimensional Lie algebras

of type A. In this case, it is well known that Littlewood-Richardson tableaux count multiplicities

of irreducible modules in the tensor product. We give a simple refinement of this rule for Kostant-Kumar modules.

This is the speaker's Ph.D defence talk.

Zoom link:

zoom.us/j/93830131758?pwd=ODJTUTZoM1NhNnJML3ZFRXZmU0Zvdz09

Meeting ID: 938 3013 1758

Passcode: 600135

Done