Semi-simple partition algebras as centralizers of representations for rook monoids
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909793079263232 |
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| author | Mazorchuk, Volodymyr Srivastava, Shraddha |
| author_facet | Mazorchuk, Volodymyr Srivastava, Shraddha |
| contents | Let $\mathcal{P}_k(δ)$, where $k$ is a positive integer and $δ$ some complex parameter, be the classical partition algebra over the complex numbers. In the case when $δ=n$, it is well-known that the algebra $\mathcal{P}_k(δ)$ is the centralizer of the symmetric group $S_n$ acting on the $k$-fold tensor space of the natural representation of $S_n$, for $n\geq 2k$. The algebra $\mathcal{P}_k(δ)$ is semi-simple for generic values of $δ$. In this paper, we show that semi-simple partition algebras appear as the centralizer algebras for certain representations of the rook monoids given by an iterative restriction-induction of the trivial representation. Along the way, we also give a decomposition of this iterative representation of the rook monoid into various tensor spaces and show that the corresponding dimensions are given by generalized Bell numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semi-simple partition algebras as centralizers of representations for rook monoids Mazorchuk, Volodymyr Srivastava, Shraddha Representation Theory Combinatorics Let $\mathcal{P}_k(δ)$, where $k$ is a positive integer and $δ$ some complex parameter, be the classical partition algebra over the complex numbers. In the case when $δ=n$, it is well-known that the algebra $\mathcal{P}_k(δ)$ is the centralizer of the symmetric group $S_n$ acting on the $k$-fold tensor space of the natural representation of $S_n$, for $n\geq 2k$. The algebra $\mathcal{P}_k(δ)$ is semi-simple for generic values of $δ$. In this paper, we show that semi-simple partition algebras appear as the centralizer algebras for certain representations of the rook monoids given by an iterative restriction-induction of the trivial representation. Along the way, we also give a decomposition of this iterative representation of the rook monoid into various tensor spaces and show that the corresponding dimensions are given by generalized Bell numbers. |
| title | Semi-simple partition algebras as centralizers of representations for rook monoids |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2509.14044 |