An imperceptible connection between the Clebsch--Gordan coefficients of $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866913882003472384 |
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| author | Huang, Hau-Wen |
| author_facet | Huang, Hau-Wen |
| contents | The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ 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 $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$. Let $Ω$ denote a finite set of size $D$ and $2^Ω$ denote the power set of $Ω$. It is generally known that $\mathbb C^{2^Ω}$ supports a $U(\mathfrak{sl}_2)$-module. Let $k$ denote an integer with $0\leq k\leq D$ and fix a $k$-element subset $x_0$ of $Ω$. By identifying $\mathbb C^{2^Ω}$ with $\mathbb C^{2^{Ω\setminus x_0}}\otimes \mathbb C^{2^{x_0}}$ this induces a $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module structure on $\mathbb C^{2^Ω}$ denoted by $\mathbb C^{2^Ω}(x_0)$. Pulling back via $\natural$ the $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module $\mathbb C^{2^Ω}(x_0)$ forms an $\mathcal H$-module. When $1\leq k\leq D-1$ the $\mathcal H$-module $\mathbb C^{2^Ω}(x_0)$ enfolds the Terwilliger algebra of the Johnson graph $J(D,k)$ with respect to $x_0$. This result connects these two seemingly irrelevant topics: The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ 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 $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs is successfully disclosed in this paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07851 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An imperceptible connection between the Clebsch--Gordan coefficients of $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs Huang, Hau-Wen Combinatorics Quantum Algebra 05E30, 06A11, 16G30, 33D80 The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ 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 $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$. Let $Ω$ denote a finite set of size $D$ and $2^Ω$ denote the power set of $Ω$. It is generally known that $\mathbb C^{2^Ω}$ supports a $U(\mathfrak{sl}_2)$-module. Let $k$ denote an integer with $0\leq k\leq D$ and fix a $k$-element subset $x_0$ of $Ω$. By identifying $\mathbb C^{2^Ω}$ with $\mathbb C^{2^{Ω\setminus x_0}}\otimes \mathbb C^{2^{x_0}}$ this induces a $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module structure on $\mathbb C^{2^Ω}$ denoted by $\mathbb C^{2^Ω}(x_0)$. Pulling back via $\natural$ the $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module $\mathbb C^{2^Ω}(x_0)$ forms an $\mathcal H$-module. When $1\leq k\leq D-1$ the $\mathcal H$-module $\mathbb C^{2^Ω}(x_0)$ enfolds the Terwilliger algebra of the Johnson graph $J(D,k)$ with respect to $x_0$. This result connects these two seemingly irrelevant topics: The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ 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 $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs is successfully disclosed in this paper. |
| title | An imperceptible connection between the Clebsch--Gordan coefficients of $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs |
| topic | Combinatorics Quantum Algebra 05E30, 06A11, 16G30, 33D80 |
| url | https://arxiv.org/abs/2308.07851 |