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

Abstract

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 𝔎ω 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 𝔎ω is the case of the Askey-Wilson algebras corresponding to the Krawtchouk polynomials. The algebra 𝔎ω is isomorphic to U(𝔰𝔩₂) when ω² ≠ 1. We view 𝑉(𝐷) as a 𝔎₁₋₂/𝑞-module. We apply the Clebsch-Gordan rule for U(𝔰𝔩₂) to decompose 𝑉(𝐷) into a direct sum of irreducible 𝒯(𝐷)-modules.