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

Abstract

Let $D\ge 1$ and $q\ge 3$ 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 $A^{2}B-2ABA+B{A}^2=B+\omega A$, $B^{2}A-2BAB+A{B}^2=A+\omega B$. 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 ${\omega }^2\ne 1$. We view $V(D)$ as a ${\mathfrak{K}}_{1-\frac{2}{q}}$-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.

References in corpus (5)

• Normalized Leonard pairs and Askey-Wilson relations
• Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
• Entanglement of Free Fermions on Johnson Graphs
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero
• Entanglement entropy in the Hamming networks