The Clebsch–Gordan coefficients ofU(𝔰𝔩₂)and the Terwilliger algebras of Johnson graphs

• Main Author: Huang, Hau-Wen
• Format: Journal Article
• Language: English
• Published: 10.12.2022
• Subjects: Mathematics - Combinatorics; Mathematics - Representation Theory
• DOI: 10.48550/arxiv.2212.05385

Journal of Combinatorial Theory, Series A, Volume 203, April 2024, 105833 The universal enveloping algebra$U(\mathfrak{sl}_2)$of$\mathfrak{sl}_2$is a unital associative algebra over$\mathbb C$generated by$E,F,H$subject to the relations align* [H,E]=2E, [H,F]=-2F, [E,F]=H. align* The element$$ \Lambda=EF+FE+\frac{H^2}{2} $$is called the Casimir element of$U(\mathfrak{sl}_2)$ . Let$\Delta:U(\mathfrak{sl}_2)\to U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$denote the comultiplication of$U(\mathfrak{sl}_2)$ . The universal Hahn algebra$\mathcal H$is a unital associative algebra over$\mathbb C$generated by$A,B,C$and the relations assert that$[A,B]=C$and each of align* [C,A]+2A^2+B, [B,C]+4BA+2C align* is central in$\mathcal H$ . Inspired by the Clebsch--Gordan coefficients of$U(\mathfrak{sl}_2)$ , we discover an algebra homomorphism$\natural:\mathcal H\to U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$that maps eqnarray* A & & H 1-1 H4, \\ B & & ()2, \\ C & & E F-F E. eqnarray* By pulling back via$\natural$any$U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$ -module can be considered as an$\mathcal H$ -module. For any integer$n\geq 0$there exists a unique$(n+1)$ -dimensional irreducible$U(\mathfrak{sl}_2)$ -module$L_n$up to isomorphism. We study the decomposition of the$\mathcal H$ -module$L_m\otimes L_n$for any integers$m,n\geq 0$ . We link these results to the Terwilliger algebras of Johnson graphs. We express the dimensions of the Terwilliger algebras of Johnson graphs in terms of binomial coefficients.