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The Clebsch–Gordan Rule for $U(\mathfrak{sl}_2)$, the Krawtchouk Algebras and the Hamming Graphs
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $D\geq 1$ and $q\geq 3$ 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 $A^2 B-2 ABA +B A^2 =B+\omega A$, $B^2A-2 BAB+AB^2=A+\omega B$. 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 $\omega^2\not=1$. We view $V(D)$ as a \smash{$\mathfrak{K}_{1-\frac{2}{q}}$}-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.
Keywords: Clebsch–Gordan rule, Hamming graph, Terwilliger algebra.
Mots-clés : Krawtchouk algebra 
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     author = {Hau-Wen Huang},
     title = {The {Clebsch{\textendash}Gordan} {Rule} for $U(\mathfrak{sl}_2)$, the {Krawtchouk} {Algebras} and the {Hamming} {Graphs}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a16/}
}
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Hau-Wen Huang. The Clebsch–Gordan Rule for $U(\mathfrak{sl}_2)$, the Krawtchouk Algebras and the Hamming Graphs. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a16/

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