Combinatorial insights on tensor power decomposition in $U_q({\mathfrak{sl}}_2)-$module

Abstract

Let $V(1)$ be the natural representation of $U({\mathfrak{sl}}_2).$ The multiplicities of $V(k)$ in $V(1{)}^{\otimes N}$ have multiple interpretations in combinatorics. In this paper, we investigate one such combinatorial interpretation of their multiplicities. Furthermore, we extend our analysis to the two-dimensional representation $T(1)$ of the restricted quantum enveloping algebra $U_q^{\mathrm{res}}({\mathfrak{sl}}_2)$. By employing combinatorial techniques, this study aims to elucidate the multiplicity of $T(k)$ within $T(1{)}^{\otimes N}$ while also offering a comprehensive analysis of the asymptotic behavior of these multiplicities as the parameter $N$ approaches infinity.

18 pages