## 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 K_{ω} defined by generators and relations. The generators are A and B. The relations are A^{2}B − 2ABA + BA^{2} = B + ωA, B^{2}A − 2BAB + AB^{2} = 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(sl_{2}) when ω^{2} ≠ 1. We view V (D) as a K_{1−} 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.

Original language | English |
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Article number | 017 |

Journal | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) |

Volume | 19 |

DOIs | |

State | Published - 2023 |

## Keywords

- Clebsch–Gordan rule
- Hamming graph
- Krawtchouk algebra
- Terwilliger algebra

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