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

Abstract

Let $\{B({\Lambda }_m)|m\in \mathbb{Z}/e\mathbb{Z}\}$ be the set of level one $\mathfrak{g}(A_{e-1}^{(1)})$-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 $B({\Lambda }_m)$.

52 pages, (v2,v3) typos and reference numbers corrected, (v4,v5) minor revision