The Terwilliger algebra of digraphs I -- Hamming digraph $H^*(d,3)$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909554431754240 |
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| author | Miezaki, Tsuyoshi Suzuki, Hiroshi Uchida, Keisuke |
| author_facet | Miezaki, Tsuyoshi Suzuki, Hiroshi Uchida, Keisuke |
| contents | In the present paper, we define the Terwilliger algebra of digraphs. Then, we determine the irreducible modules of the Terwilliger algebra of a Hamming digraph $H^*(d,3)$. As is well known, the representation of the Terwilliger algebra of a binary Hamming graph $H(d,2)$ is closely related to that of the Lie algebra $\mathit{sl}_2(\mathbb{C})$. We show that in the case of $H^*(d,3)$, it is related to that of the Lie algebra $\mathit{sl}_3(\mathbb{C})$. We also identify the Terwilliger algebra of $H^*(d,3)$ as the $d$ symmetric tensor algebra of ${\rm Mat}_3(\mathbb{C})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21077 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Terwilliger algebra of digraphs I -- Hamming digraph $H^*(d,3)$ Miezaki, Tsuyoshi Suzuki, Hiroshi Uchida, Keisuke Combinatorics Representation Theory Primary 05C25, Secondary 17B10, 05E30 In the present paper, we define the Terwilliger algebra of digraphs. Then, we determine the irreducible modules of the Terwilliger algebra of a Hamming digraph $H^*(d,3)$. As is well known, the representation of the Terwilliger algebra of a binary Hamming graph $H(d,2)$ is closely related to that of the Lie algebra $\mathit{sl}_2(\mathbb{C})$. We show that in the case of $H^*(d,3)$, it is related to that of the Lie algebra $\mathit{sl}_3(\mathbb{C})$. We also identify the Terwilliger algebra of $H^*(d,3)$ as the $d$ symmetric tensor algebra of ${\rm Mat}_3(\mathbb{C})$. |
| title | The Terwilliger algebra of digraphs I -- Hamming digraph $H^*(d,3)$ |
| topic | Combinatorics Representation Theory Primary 05C25, Secondary 17B10, 05E30 |
| url | https://arxiv.org/abs/2503.21077 |