The Clebsch-Gordan rule for \(U(\mathfrak{sl}_2)\), the Krawtchouk algebras and the Hamming graphs

DOI10.3842/SIGMA.2023.017zbMATH Open1512.05409arXiv2106.06857OpenAlexW4362561447MaRDI QIDQ6042952

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Publication date: 4 May 2023

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Abstract: Let $Dgeq1$ and $qgeq3$ be two integers. Let $H(D)=H(D,q)$ denote the $D$-dimensional Hamming graph over a $q$-element set. Let $mathcalT(D)$ denote the Terwilliger algebra of $H(D)$. Let $V(D)$ denote the standard $mathcalT(D)$-module. Let $omega$ denote a complex scalar. We consider a unital associative algebra $mathfrak{K}_{o}mega$ defined by generators and relations. The generators are $A$ and $B$. The relations are $A^{2}B-2ABA+B{A}^2=B+omegaA$, $B^{2}A-2BAB+A{B}^2=A+omegaB$. The algebra $mathfrak{K}_{o}mega$ is the case of the Askey-Wilson algebras corresponding to the Krawtchouk polynomials. The algebra $mathfrak{K}_{o}mega$ is isomorphic to $mU(mathfrak{sl}_2)$ when $omeg{a}^{2}ot=1$. We view $V(D)$ as a $mathfrak{K}_{1-frac2q}$-module. We apply the Clebsch-Gordan rule for $mU(mathfrak{sl}_2)$ to decompose $V(D)$ into a direct sum of irreducible $mathcalT(D)$-modules.

Full work available at URL: https://arxiv.org/abs/2106.06857

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