The Clebsch–Gordan Rule for U(sl₂), the Krawtchouk Algebras and the Hamming Graphs

Let D ≥ 1 and q ≥ 3 be two integers. Let H(D) = H(D, q) denote the D-dimensional Hamming graph over a q-element set. Let T (D) denote the Terwilliger algebra of H(D). Let V (D) denote the standard T (D)-module. Let ω denote a complex scalar. We consider a unital associative algebra K_ω defined by generators and relations. The generators are A and B. The relations are A²B − 2ABA + BA² = B + ωA, B²A − 2BAB + AB² = A + ωB. The algebra K_ω is the case of the Askey–Wilson algebras corresponding to the Krawtchouk polynomials. The algebra K_ω is isomorphic to U(sl₂) when ω² ≠ 1. We view V (D) as a K₁₋ 2^{-module. We apply the Clebsch–Gordan rule for U(sl}2) q to decompose V (D) into a direct sum of irreducible T (D)-modules.