An imperceptible connection between the Clebsch--Gordan coefficients of $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs

Abstract

The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism $\natural$ from the universal Hahn algebra $\mathcal H$ into $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$. Let $\Omega$ denote a finite set of size $D$ and $2^\Omega$ denote the power set of $\Omega$. It is generally known that $\mathbb C^{2^\Omega}$ supports a $U(\mathfrak{sl}_2)$-module. Let $k$ denote an integer with $0\leq k\leq D$ and fix a $k$-element subset $x_0$ of $\Omega$. By identifying $\mathbb C^{2^\Omega}$ with $\mathbb C^{2^{\Omega\setminus x_0}}\otimes \mathbb C^{2^{x_0}}$ this induces a $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module structure on $\mathbb C^{2^\Omega}$ denoted by $\mathbb C^{2^\Omega}(x_0)$. Pulling back via $\natural$ the $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module $\mathbb C^{2^\Omega}(x_0)$ forms an $\mathcal H$-module. When $1\leq k\leq D-1$ the $\mathcal H$-module $\mathbb C^{2^\Omega}(x_0)$ 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 $U(\mathfrak{sl}_2)$ 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 $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs is successfully disclosed in this paper.