Consider the tensor product of two irreducible finite dimensional representations of a simple Lie algebra. The submodules generated by a tensor product of extremal vectors of the two components are called Kostant-Kumar submodules. These are parametrized by a double coset space of the Weyl group.

Littelmannâ€™s path model is a very general combinatorial model for representations, which encompasses many classical constructs such as Young tableaux and Lakshmibai-Seshadri chains. The path model for the full tensor product is simply the set of concatenations of paths of the individual components. We describe a way to associate a Weyl group element (rather, a double coset) to each such concatenated path and thereby obtain a path model for Kostant-Kumar submodules.

Finally, we recall the many descriptions of Demazure modules, which may be viewed as the analog of the above picture for single paths (rather than concatenations). In this case, the Weyl group element associated to a path admits different descriptions in different path models in terms of statistics such as initial direction, minimal standard lifts and Kogan faces of Gelfand-Tsetlin polytopes. Along the way, we mention some relations to jeu-de-taquin and the Schutzenberger involution.

This is based on joint work with Mrigendra Singh Kushwaha and KN Raghavan.

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Last updated: 15 Jul 2024