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

Let $\{B({\mathrm{\Lambda }}_m)\mathrm{\mid }m\in \mathbb{Z}\mathrm{/}e\mathbb{Z}\}$ be the set of level one $\mathfrak{g}(A_{e-1}^{(1)})$-crystals, and consider the realization of $B({\mathrm{\Lambda }}_m)$ using $e$-restricted partitions. We prove a purely Young diagrammatic criterion for an element of $B(\Lambda_0)^{\otimes d_1}\otimes B(\Lambda_m)^{\otimes d_2}$tobeinthecomponent$B(d_1\Lambda_0+d_2\Lambda_m)$. 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 $B({\mathrm{\Lambda }}_m)$.