Higher-spin quantum and classical Schur-Weyl duality for $\mathfrak{sl}_2$

Abstract

It is well-known that the commutant algebra of the $U_q(\mathfrak{sl}_2)$-action on the $n$-fold tensor product of its fundamental module is isomorphic to the Temperley-Lieb algebra TL$_n(ν)$ with fugacity parameter $ν= -q - q^{-1}$ (at least in the generic case, i.e., when $q$ is not a root of unity, or $n$ is small enough). Furthermore, the simple $U_q(\mathfrak{sl}_2)$-modules appearing in the direct-sum decomposition of the $n$-fold tensor product module are in one-to-one correspondence with those of the Temperley-Lieb algebra. This double-commutant property is referred to as quantum Schur-Weyl duality. In this article, we investigate such a duality in great detail. We prove that the commutant of the $U_q(\mathfrak{sl}_2)$-action on any generic type-one tensor product module is isomorphic to a diagram algebra that we call the valenced Temperley-Lieb algebra TL$_ς(ν)$. This corresponds to representations with higher spin, which results in the need of valences (or colors) in the Temperley-Lieb diagrams. We establish detailed direct-sum decompositions exhibiting this duality and find explicit bases amenable to concrete calculations, important in applications. We also include a double-commutant type property for homomorphisms between different $U_q(\mathfrak{sl}_2)$-modules, realized by valenced diagrams. The diagram calculus is reminiscent to Kauffman's recoupling theory and the graphical methods developed among others by Penrose and Frenkel \& Khovanov. The results also contain the standard quantum Schur-Weyl duality as a special case, and when specialized to $q \rightarrow 1$, imply the classical Frobenius-Schur-Weyl duality for the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ and a higher-spin version thereof.