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

Hau-Wen Huang

Published 2023-08-15Version 1

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 from the universal Hahn algebra $\mathcal H$ into $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$. Let $\Omega$ denote a finite set 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. Fix an element $x_0\in 2^\Omega$. By the linear isomorphism $\mathbb C^{2^\Omega}\to \mathbb C^{2^{\Omega\setminus x_0}}\otimes \mathbb C^{2^{x_0}}$ given by $x\mapsto (x\setminus x_0)\otimes (x\cap x_0)$ for all $x\in 2^\Omega$, this induces a $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module structure on $\mathbb C^{2^\Omega}$. Pulling back via the algebra homomorphism $\mathcal H\to U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$, the $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module $\mathbb C^{2^\Omega}$ forms an $\mathcal H$-module. The $\mathcal H$-module $\mathbb C^{2^\Omega}$ enfolds the Terwilliger algebra of a Johnson graph. 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.