2023/08/15 by Hau-Wen Huang, Huang, Hau-Wen · 3 citations
Mathematics · #05E30 #06A11 #16G30 #33D80 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2308.07851
openalex publication_date 2023/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Clebsch--Gordan coefficients of U(\mathfraksl2) are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism \natural from the universal Hahn algebra \mathcal H into U(\mathfraksl2)⊗ U(\mathfraksl2). Let Ω denote a finite set of size D and 2Ω denote the power set of Ω. It is generally known that \mathbb C2Ω supports a U(\mathfraksl2)-module. Let k denote an integer with 0≤ k≤ D and fix a k-element subset x0 of Ω. By identifying \mathbb C2Ω with \mathbb C^2Ω∖ x0⊗ \mathbb C^2x0 this induces a U(\mathfraksl2)⊗ U(\mathfraksl2)-module structure on \mathbb C2Ω denoted by \mathbb C2Ω(x0). Pulling back via \natural the U(\mathfraksl2)⊗ U(\mathfraksl2)-module \mathbb C2Ω(x0) forms an \mathcal H-module. When 1≤ k≤ D-1 the \mathcal H-module \mathbb C2Ω(x0) enfolds the Terwilliger algebra of the Johnson graph J(D,k) with respect to x0. This result connects these two seemingly irrelevant topics: The Clebsch--Gordan coefficients of U(\mathfraksl2) and the Terwilliger algebras of Johnson graphs. Unfortunately some steps break down in the q-analog case. By making detours, the imperceptible connection between the Clebsch--Gordan coefficients of Uq(\mathfraksl2) and the Terwilliger algebras of Grassmann graphs is successfully disclosed in this paper.