Speaker: Patrick Polo
Affiliation: Sorbonne University and Chennai Mathematical Institute
Title of Talk: An introduction to Kashiwara's crystals and components of $L(\rho)\otimes L(\rho)$ associated with rooted trees for symmetrizable Kac-Moody algebras
Date: August 27, 2026
Time: 11:00:00 Hours
Venue: A-369

Abstract: Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra over $\mathbb{C}$. Let $L(\lambda)$ denote the integrable $\mathfrak{g}$-module with highest weight $\lambda$, for every dominant integral weight $\lambda$. Each tensor product $L(\lambda)\otimes L(\mu)$ decomposes as a direct sum $\bigoplus_{\nu \leq \lambda + \mu} L(\nu)^{c_{\lambda,\mu}^\nu}$, for some $c_{\lambda,\mu}^\nu \in \mathbb{N}$. We will explain Kashiwara's theory of crystals, which gives that $c_{\lambda,\mu}^\nu$ is the number of $\mu$-dominant element of weight $\nu-\lambda$ in the crystal $\mathcal{B}(\lambda)$ (a combinatorial version of a result by Parthasarathy, Ranga Rao and Varadarajan). Then we will specialize to the case where $\lambda = \mu = \rho$ and $2\rho-\nu$ is a sum of simple roots with coefficients $\leq 2$, whose support is a subgraph of the Dynkin diagram of $\mathfrak{g}$ which has only simple bonds and no cycle of length $\geq 3$. In this case, using some combinatorics about rooted trees, we construct explicitly a $\rho$-dominant element of weight $\nu-\rho$ in $\mathcal{B}(\rho)$. This is a small step towards a conjecture of Kostant. (This talk is based on joint work with Rekha Biswal (NISER Bhubaneswar).)