The Clebsch-Gordan Rule for U(sl<sub>2</sub>), the Krawtchouk Algebras and the Hamming Graphs

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