The Clebsch–Gordan Rule for $U(\mathfrak{sl}_2)$, the Krawtchouk Algebras and the Hamming Graphs

Let $D\geq 1$ and $q\geq 3$ be two integers. Let $H(D)=H(D,q)$ denote the $D$-dimensional Hamming graph over a $q$-element set. Let $\mathcal{T}(D)$ denote the Terwilliger algebra of $H(D)$. Let $V(D)$ denote the standard $\mathcal{T}(D)$-module. Let $\omega$ denote a complex scalar. We consider a unital associative algebra $\mathfrak{K}_\omega$ defined by generators and relations. The generators are $A$ and $B$. The relations are $A^2 B-2 ABA +B A^2 =B+\omega A$, $B^2A-2 BAB+AB^2=A+\omega B$. The algebra $\mathfrak{K}_\omega$ is the case of the Askey–Wilson algebras corresponding to the Krawtchouk polynomials. The algebra $\mathfrak{K}_\omega$ is isomorphic to $\mathrm{U}({\mathfrak{sl}_2)}$ when $\omega^2\not=1$. We view $V(D)$ as a \smash{$\mathfrak{K}_{1-\frac{2}{q}}$}-module. We apply the Clebsch–Gordan rule for $\mathrm{U}({\mathfrak{sl}_2)}$ to decompose $V(D)$ into a direct sum of irreducible $\mathcal{T}(D)$-modules.