Theoretical Statistics and Mathematics Unit, ISI Delhi
The occurrence of certain natural irreducible modules as summands in the decomposition of the tensor product of two irreducible representations of a complex semisimple Lie algebra was conjectured by Parthasarathy, Ranga Rao, and Varadarajan in the nineteen-sixties. The conjecture was proved in the nineteen-eighties, independently by Kumar and Mathieu. In the early nineteen-nineties, Littelmann deduced the conjecture from his version of the Littlewood-Richardson rule which gives a description of the decomposition as a direct sum of irreducibles of the tensor product.
Littelmann's proof is based on his theory of the path model, which he developed building on the work of Lakshmibai and Seshadri on "Standard Monomial Theory". The path model is a combinatorial gadget that is a far reaching generalization of the more classical notion of semi-standard Young tableaux (whose relevance for representation theory has been known since the first quarter of the twentieth century).
Kumar's proof of the PRV conjecture is based on a refinement of it due to Kostant: there is a natural filtration, indexed by the Weyl group, by "Kostant modules" of the tensor product.
The present talk is a report on joint work with Mrigendra Singh Kushwaha and Sankaran Viswanath where we approach the Kostant version of the PRV statement via the path model. We prove a Kostant refinement of the Littlewood-Richardson rule, identity path models for the Kostant modules, and give generalizations and refinements of the Kostant version of the PRV theorem.
An attempt will be made to render the talk accessible to a wide mathematical audience.