On the Terwilliger algebra of the group association scheme of the symmetric group $\operatorname {sym}(7)$
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| Format: | Preprint |
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2024
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| _version_ | 1866915018280271872 |
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| author | Herman, Allen Maleki, Roghayeh Razafimahatratra, Andriaherimanana Sarobidy |
| author_facet | Herman, Allen Maleki, Roghayeh Razafimahatratra, Andriaherimanana Sarobidy |
| contents | Terwilliger algebras are finite-dimensional semisimple algebras that were first introduced by Paul Terwilliger in 1992 in studies of association schemes and distance-regular graphs. The Terwilliger algebras of the conjugacy class association schemes of the symmetric groups $\operatorname {sym}(n)$, for $3\leq n \leq 6$, have been studied and completely determined. The case for $\operatorname {sym}(7)$ is computationally much more difficult and has a potential application to find the size of the largest permutation codes of $\operatorname {sym}(7)$ with a minimal distance of at least $4$. In this paper, the dimension, the Wedderburn decomposition, and the block dimension decomposition of the Terwilliger algebra of the conjugacy class scheme of the group $\operatorname {sym}(7)$ are determined. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_08803 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Terwilliger algebra of the group association scheme of the symmetric group $\operatorname {sym}(7)$ Herman, Allen Maleki, Roghayeh Razafimahatratra, Andriaherimanana Sarobidy Combinatorics Rings and Algebras 05E30, 16Z05 Terwilliger algebras are finite-dimensional semisimple algebras that were first introduced by Paul Terwilliger in 1992 in studies of association schemes and distance-regular graphs. The Terwilliger algebras of the conjugacy class association schemes of the symmetric groups $\operatorname {sym}(n)$, for $3\leq n \leq 6$, have been studied and completely determined. The case for $\operatorname {sym}(7)$ is computationally much more difficult and has a potential application to find the size of the largest permutation codes of $\operatorname {sym}(7)$ with a minimal distance of at least $4$. In this paper, the dimension, the Wedderburn decomposition, and the block dimension decomposition of the Terwilliger algebra of the conjugacy class scheme of the group $\operatorname {sym}(7)$ are determined. |
| title | On the Terwilliger algebra of the group association scheme of the symmetric group $\operatorname {sym}(7)$ |
| topic | Combinatorics Rings and Algebras 05E30, 16Z05 |
| url | https://arxiv.org/abs/2411.08803 |