On the tensor product of two basic representations of $U_v(\hat{sl}_e)$

Abstract

Let $\{B(\Lambda_m)|m\in\Z/e\Z\}$ be the set of level one $\mathfrak{g}(A^{(1)}_{e-1})$-crystals, and consider the realization of $B(\Lambda_m)$ using $e$-restricted partitions. We prove a purely Young diagrammatic criterion for an element of $B(\Lambda_0)^{\otimes d_1}\otimes B(\Lambda_m)^{\otimes d_2}$ to be in the component $B(d_1\Lambda_0+d_2\Lambda_m)$. As an application, we give a non-recursive characterization of simple modules of the Hecke algebra of type $B$. In the course of the proof, we also obtain a combinatorial description of the second type of Kashiwara's Demazure crystal in $B(\Lambda_m)$.