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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2507.14313 |
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| _version_ | 1866915400467349504 |
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| author | Campbell, John M. |
| author_facet | Campbell, John M. |
| contents | Wilcox has considered a twisted semigroup algebra structure on the partition algebra $\mathbb{C}A_k(n)$, but it appears that there has not previously been any known basis that gives $\mathbb{C}A_k(n)$ the structure of a "non-twisted" semigroup algebra or a monoid algebra. This motivates the following problem, for the non-degenerate case whereby $n \in \mathbb{C} \setminus \{ 0, 1, \ldots, 2 k - 2 \}$ so that $ \mathbb{C}A_k(n)$ is semisimple. How could a basis $M_{k} = M$ of $ \mathbb{C}A_k(n)$ be constructed so that $M$ is closed under the multiplicative operation on $\mathbb{C}A_k(n)$, in such a way so that $M$ is a monoid under this operation, and how could a product rule for elements in $M$ be defined in an explicit and combinatorial way in terms of partition diagrams? We construct a basis $M$ of the desired form using Halverson and Ram's matrix unit construction for partition algebras, Benkart and Halverson's bijection between vacillating tableaux and set-partition tableaux, an analogue given by Colmenarejo et al. for partition diagrams of the RSK correspondence, and a variant of a result due to Hewitt and Zuckerman characterizing finite-dimensional semisimple algebras that are isomorphic to semigroup algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14313 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partition algebras as monoid algebras Campbell, John M. Combinatorics 05E10 Wilcox has considered a twisted semigroup algebra structure on the partition algebra $\mathbb{C}A_k(n)$, but it appears that there has not previously been any known basis that gives $\mathbb{C}A_k(n)$ the structure of a "non-twisted" semigroup algebra or a monoid algebra. This motivates the following problem, for the non-degenerate case whereby $n \in \mathbb{C} \setminus \{ 0, 1, \ldots, 2 k - 2 \}$ so that $ \mathbb{C}A_k(n)$ is semisimple. How could a basis $M_{k} = M$ of $ \mathbb{C}A_k(n)$ be constructed so that $M$ is closed under the multiplicative operation on $\mathbb{C}A_k(n)$, in such a way so that $M$ is a monoid under this operation, and how could a product rule for elements in $M$ be defined in an explicit and combinatorial way in terms of partition diagrams? We construct a basis $M$ of the desired form using Halverson and Ram's matrix unit construction for partition algebras, Benkart and Halverson's bijection between vacillating tableaux and set-partition tableaux, an analogue given by Colmenarejo et al. for partition diagrams of the RSK correspondence, and a variant of a result due to Hewitt and Zuckerman characterizing finite-dimensional semisimple algebras that are isomorphic to semigroup algebras. |
| title | Partition algebras as monoid algebras |
| topic | Combinatorics 05E10 |
| url | https://arxiv.org/abs/2507.14313 |