Upper bounds for measures on distal classes
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866929439007309824 |
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| author | Nekrasov, Ilia Snowden, Andrew |
| author_facet | Nekrasov, Ilia Snowden, Andrew |
| contents | In recent work, Harman and Snowden introduced a notion of measure on a Fraïssé class $\mathfrak{F}$, and showed how such measures lead to interesting tensor categories. Constructing and classifying measures is a difficult problem, and so far only a handful of cases have been worked out. In this paper, we obtain some of the first general results on measures. Our main theorem states that if $\mathfrak{F}$ is distal (in the sense of Simon), and there are some bounds on automorphism groups, then $\mathfrak{F}$ admits only finitely many measures; moreover, we give an effective upper bound on their number. For example, if $\mathfrak{F}$ is the class of ``$s$-dimensional permutations'' (finite sets equipped with $s$ total orders), we show that the number of measures is bounded above by approximately $\exp(\exp(s^2 \log{s}))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19131 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper bounds for measures on distal classes Nekrasov, Ilia Snowden, Andrew Representation Theory Combinatorics Logic In recent work, Harman and Snowden introduced a notion of measure on a Fraïssé class $\mathfrak{F}$, and showed how such measures lead to interesting tensor categories. Constructing and classifying measures is a difficult problem, and so far only a handful of cases have been worked out. In this paper, we obtain some of the first general results on measures. Our main theorem states that if $\mathfrak{F}$ is distal (in the sense of Simon), and there are some bounds on automorphism groups, then $\mathfrak{F}$ admits only finitely many measures; moreover, we give an effective upper bound on their number. For example, if $\mathfrak{F}$ is the class of ``$s$-dimensional permutations'' (finite sets equipped with $s$ total orders), we show that the number of measures is bounded above by approximately $\exp(\exp(s^2 \log{s}))$. |
| title | Upper bounds for measures on distal classes |
| topic | Representation Theory Combinatorics Logic |
| url | https://arxiv.org/abs/2407.19131 |