3:30 pm, Seminar Hall
A study of Kostant-Kumar modules via Littelmann paths
K.N. Raghavan
Institute of Mathematical Sciences, Chennai.
10-04-19

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Abstract

Consider finite dimensional representations of a complex semisimple Lie algebra (or such a Lie group / algebraic group / or even a symmetrizable Kac-Moody algebra). Into this representation theory, Littelmann introduced the notion of paths, which is a far reaching generalization of the classical notion of tableaux. Like tableaux, paths form a combinatorial proxy for representations. Operations on representations tend to leave discernible shadows on paths.

Recall that irreducible representations are indexed by dominant integral weights. Given such a weight l, the set of Lakshmibai-Seshadri paths (LS paths) of shape l forms a combinatorial shadow of the corresponding irreducible representation. The tensor product operation on representations leaves concatenation as its shadow on paths. Thus, given two dominant integral weights l and m, the set of concatenations of LS paths of shapes l and m forms a "path model" for the tensor product of the corresponding two irreducible representations. Using this, Littelmann derived a rule ("Littelwood-Richardson rule") for how the tensor product decomposes as a direct sum of irreducible representations.

Now consider the Kostant-Kumar (KK) submodules of a tensor product of two irreducible representations. These are cyclic submodules generated by the highest weight vector tensor an extremal weight vector. They form a filtration indexed by the Weyl group of the tensor product. One can ask what shadow this filtration leaves on paths.

We describe a way to assign a Weyl group element to a concatenation of two LS paths. Our main technical result is that this assignment doesn't change when the concatenated path is acted upon by root operators. We observe that the resulting filtration (indexed by the Weyl group) on the set of concatenated LS paths is a faithful shadow of the KK filtration on the tensor product. In other words, we obtain a path model for KK submodules. Further, we obtain a decomposition rule for how a KK submodule breaks up as a direct sum of irreducibles. Using these results we can prove generalizations of "PRV type" results of Kumar and others. Our results and proofs are modeled after those for the tensor product by Littelmann.

This is joint work with Mrigendra Singh Kushwaha and Sankaran Viswanath (both of IMSc). The talk will be accessible to anybody who knows what the character of a representation is.