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

TL;DR

This paper provides a combinatorial framework for understanding crystal bases of extremal weight modules over quantum infinite rank affine algebras of types B, C, and D, and describes tensor product decompositions explicitly.

Contribution

It extends previous results by offering a combinatorial realization and explicit tensor product decomposition formulas for these crystal bases in types B, C, and D.

Findings

Crystal bases realized via Lakshmibai-Seshadri paths.

Explicit tensor product decomposition in terms of Littlewood-Richardson coefficients.

Extension of known results from types A to types B, C, and D.

Abstract

Using Lakshmibai-Seshadri paths, we give a combinatorial realization of the crystal basis of an extremal weight module of 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.