Classification of indecomposable states on the infinite symmetric inverse semigroup invariant under the infinite symmetric group. Semifinite case
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913997776748544 |
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| author | Dudko, Artem Nessonov, Nikolay I. |
| author_facet | Dudko, Artem Nessonov, Nikolay I. |
| contents | Let $\mathbb{N}$ be a set of the natural numbers.
Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $[g_{ij}]$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a finite set. The binary operation in $R_\infty$ is the ordinary matrix multiplication. It is clear that infinite symmetric group $\mathfrak{S}_\infty$ is a subgroup of $R_\infty$. The map $\star:\left[ g_{ij}\right]\mapsto\left[ g_{ji}\right]$ is an involution on $R_\infty$. We call a function $f$ on $R_\infty$ positive definite if for all $r_1, r_2, \ldots, r_n\in R_\infty$ the matrix $\left[ f\left( r_ir_j^\star\right)\right]$ is Hermitian and positive semi-definite. A function $f$ said to be indecomposable if the corresponding $\ast$-representation $π_f$ is a factor-representation. A class of the $\mathfrak{S}_\infty$-invariant functions is defined by the condition $f(rs)=f(sr)$ for all $r\in R_\infty$ and $s\in\mathfrak{S}_\infty$. In this paper we classify all semifinite factor-representations of $R_\infty$ that correspond to the $\mathfrak{S}_\infty$-invariant positive definite functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13760 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of indecomposable states on the infinite symmetric inverse semigroup invariant under the infinite symmetric group. Semifinite case Dudko, Artem Nessonov, Nikolay I. Representation Theory Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $[g_{ij}]$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a finite set. The binary operation in $R_\infty$ is the ordinary matrix multiplication. It is clear that infinite symmetric group $\mathfrak{S}_\infty$ is a subgroup of $R_\infty$. The map $\star:\left[ g_{ij}\right]\mapsto\left[ g_{ji}\right]$ is an involution on $R_\infty$. We call a function $f$ on $R_\infty$ positive definite if for all $r_1, r_2, \ldots, r_n\in R_\infty$ the matrix $\left[ f\left( r_ir_j^\star\right)\right]$ is Hermitian and positive semi-definite. A function $f$ said to be indecomposable if the corresponding $\ast$-representation $π_f$ is a factor-representation. A class of the $\mathfrak{S}_\infty$-invariant functions is defined by the condition $f(rs)=f(sr)$ for all $r\in R_\infty$ and $s\in\mathfrak{S}_\infty$. In this paper we classify all semifinite factor-representations of $R_\infty$ that correspond to the $\mathfrak{S}_\infty$-invariant positive definite functions. |
| title | Classification of indecomposable states on the infinite symmetric inverse semigroup invariant under the infinite symmetric group. Semifinite case |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2508.13760 |