An imperceptible connection between the Clebsch–Gordan coefficients of Uq(sl2) and the Terwilliger algebras of Grassmann graphs

Abstract

The Clebsch–Gordan coefficients of $U\left({\mathrm{sl}}_2\right)$ are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism ♮ from the universal Hahn algebra $H$ into $U\left({\mathrm{sl}}_2\right)\otimes U\left({\mathrm{sl}}_2\right)$. Let Ω denote a finite set of size D and $2^{\Omega }$ denote the power set of Ω. It is generally known that $C^{2^{\Omega }}$ supports a $U\left({\mathrm{sl}}_2\right)$-module. Let k denote an integer with $0\le k\le D$ and fix a k-element subset $x_0$ of Ω. By identifying $C^{2^{\Omega }}$ with $C^{2^{\Omega \setminus x_0}}\otimes C^{2^{x_0}}$ this induces a $U\left({\mathrm{sl}}_2\right)\otimes U\left({\mathrm{sl}}_2\right)$-module structure on $C^{2^{\Omega }}$ denoted by $C^{2^{\Omega }}\left(x_0\right)$. Pulling back via ♮ the $U\left({\mathrm{sl}}_2\right)\otimes U\left({\mathrm{sl}}_2\right)$-module $C^{2^{\Omega }}\left(x_0\right)$ forms an $H$-module. When $1\le k\le D-1$ the $H$-module $C^{2^{\Omega }}\left(x_0\right)$ enfolds the Terwilliger algebra of the Johnson graph $J(D,k)$ with respect to $x_0$. This result connects these two seemingly irrelevant topics: The Clebsch–Gordan coefficients of $U\left({\mathrm{sl}}_2\right)$ and the Terwilliger algebras of Johnson graphs. Unfortunately some steps break down in the q-analog case. By making detours, the imperceptible connection between the Clebsch–Gordan coefficients of $U_q\left({\mathrm{sl}}_2\right)$ and the Terwilliger algebras of Grassmann graphs is successfully disclosed in this paper.