On some components of $L(\rho )\otimes L(\rho )$ associated with rooted trees for symmetrizable Kac-Moody algebras

Abstract

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra over $\mathbb{C}$ and let $L(\rho )$ be the irreducible integrable $\mathfrak{g}$-module with highest weight $\rho$. Let $I$ be a subgraph of the Dynkin diagram of $\mathfrak{g}$ which has only simple bonds and no cycle of length $\ge 3$. For every subset $D$ of $I$, denote by ${\beta }_D$ the sum of the simple roots corresponding to $D$. To every $D\subset I$ such that ${\lambda }_{D,I}=2\rho -{\beta }_I-{\beta }_D$ is dominant, we associate certain elements ${\pi }_{D,I}$ of weight ${\lambda }_{D,I}-\rho$ in the crystal $B(\rho )$, which depend on the choice of a root vertex in each connected component of $I$. Then we prove that our elements are $\rho$-dominant elements of $B(\rho )$, hence provide new families of components of the tensor product $L(\rho )\otimes L(\rho )$.

version 2 : numbering of vertices in Example 3.18 corrected