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