Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of types $B_{\infty }$, $C_{\infty }$, and $D_{\infty }$

Abstract:

Using Lakshmibai-Seshadri paths, we give a combinatorial realization of the crystal basis of an extremal weight module of a (general) integral extremal weight over the quantized universal enveloping algebra associated to the infinite rank affine Lie algebra of type $B_{\infty }$, $C_{\infty }$, or $D_{\infty }$. Moreover, via this realization, we obtain an explicit description (in terms of Littlewood-Richardson coefficients) of how tensor products of these crystal bases decompose into connected components when their extremal weights are of nonnegative levels. These results in types $B_{\infty }$, $C_{\infty }$, and $D_{\infty }$ extend the corresponding results due to Kwon in types $A_{+\infty }$ and $A_{\infty }$. Our results above also include, as a special case, the corresponding results (concerning crystal bases) due to Lecouvey in types $B_{\infty }$, $C_{\infty }$, and $D_{\infty }$, where the extremal weights are of level zero.