The $S_3$-symmetric tridiagonal algebra
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917359843803136 |
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| author | Terwilliger, Paul |
| author_facet | Terwilliger, Paul |
| contents | The tridiagonal algebra is defined by two generators and two relations, called the tridiagonal relations. Special cases of the tridiagonal algebra include the $q$-Onsager algebra, the positive part of the $q$-deformed enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$, and the enveloping algebra of the Onsager Lie algebra.
In this paper, we introduce the $S_3$-symmetric tridiagonal algebra. This algebra has six generators. The generators can be identified with the vertices of a regular hexagon, such that nonadjacent generators commute and adjacent generators satisfy a pair of tridiagonal relations. For a $Q$-polynomial distance-regular graph $Γ$ we turn the tensor power $V^{\otimes 3}$ of the standard module $V$ into a module for an $S_3$-symmetric tridiagonal algebra.
We investigate in detail the case in which $Γ$ is a Hamming graph. We give some conjectures and open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00551 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $S_3$-symmetric tridiagonal algebra Terwilliger, Paul Combinatorics Quantum Algebra 05E30 The tridiagonal algebra is defined by two generators and two relations, called the tridiagonal relations. Special cases of the tridiagonal algebra include the $q$-Onsager algebra, the positive part of the $q$-deformed enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$, and the enveloping algebra of the Onsager Lie algebra. In this paper, we introduce the $S_3$-symmetric tridiagonal algebra. This algebra has six generators. The generators can be identified with the vertices of a regular hexagon, such that nonadjacent generators commute and adjacent generators satisfy a pair of tridiagonal relations. For a $Q$-polynomial distance-regular graph $Γ$ we turn the tensor power $V^{\otimes 3}$ of the standard module $V$ into a module for an $S_3$-symmetric tridiagonal algebra. We investigate in detail the case in which $Γ$ is a Hamming graph. We give some conjectures and open problems. |
| title | The $S_3$-symmetric tridiagonal algebra |
| topic | Combinatorics Quantum Algebra 05E30 |
| url | https://arxiv.org/abs/2407.00551 |