تقرير
An imperceptible connection between the Clebsch--Gordan coefficients of $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs
| العنوان: | 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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| المؤلفون: | Huang, Hau-Wen |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, Mathematics - Quantum Algebra, 05E30, 06A11, 16G30, 33D80 |
| الوصف: | 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 $\Omega$ denote a finite set of size $D$ and $2^\Omega$ denote the power set of $\Omega$. It is generally known that $\mathbb C^{2^\Omega}$ 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 $\Omega$. By identifying $\mathbb C^{2^\Omega}$ with $\mathbb C^{2^{\Omega\setminus x_0}}\otimes \mathbb C^{2^{x_0}}$ this induces a $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module structure on $\mathbb C^{2^\Omega}$ denoted by $\mathbb C^{2^\Omega}(x_0)$. Pulling back via $\natural$ the $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module $\mathbb C^{2^\Omega}(x_0)$ forms an $\mathcal H$-module. When $1\leq k\leq D-1$ the $\mathcal H$-module $\mathbb C^{2^\Omega}(x_0)$ 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. Comment: 65 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2308.07851 |
| رقم الأكسشن: | edsarx.2308.07851 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |