Essential dimension of symmetric groups in prime characteristic
The essential dimension $\operatorname{ed}_k(\operatorname{S}_n)$ of the symmetric group $\operatorname{S}_n$ is the minimal integer $d$ such that the general polynomial $x^n + a_1 x^{n-1} + \dots + a_n$ can be reduced to a $d$-parameter form by a Tschirnhaus transformation. Finding this number is a...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-07-01
|
| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.577/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1825206268177940480 |
|---|---|
| author | Edens, Oakley Reichstein, Zinovy |
| author_facet | Edens, Oakley Reichstein, Zinovy |
| author_sort | Edens, Oakley |
| collection | DOAJ |
| description | The essential dimension $\operatorname{ed}_k(\operatorname{S}_n)$ of the symmetric group $\operatorname{S}_n$ is the minimal integer $d$ such that the general polynomial $x^n + a_1 x^{n-1} + \dots + a_n$ can be reduced to a $d$-parameter form by a Tschirnhaus transformation. Finding this number is a long-standing open problem, originating in the work of Felix Klein, long before essential dimension of a finite group was formally defined. We now know that $\operatorname{ed}_k(\operatorname{S}_n)$ lies between $\lfloor n/2 \rfloor $ and $n-3$ for each $n \geqslant 5$ and any field $k$ of characteristic different from $2$. Moreover, if $\operatorname{char}(k) = 0$, then $\operatorname{ed}_k(\operatorname{S}_n) \geqslant \lfloor (n+1)/2 \rfloor $ for any $n \geqslant 7$. The value of $\operatorname{ed}_k(\operatorname{S}_n)$ is not known for any $n \geqslant 8$ and any field $k$, though it is widely believed that $\operatorname{ed}_k(\operatorname{S}_n)$ should be $n-3$ for every $n \geqslant 5$, at least in characteristic $0$. In this paper we show that for every prime $p$ there are infinitely many positive integers $n$ such that $\operatorname{ed}_{\mathbb{F}_p}(\operatorname{S}_n) \leqslant n-4$. |
| format | Article |
| id | doaj-art-d6d76f42301c44c0a03da648c3d99e61 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-07-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-d6d76f42301c44c0a03da648c3d99e612025-02-07T11:21:51ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-07-01362G663964710.5802/crmath.57710.5802/crmath.577Essential dimension of symmetric groups in prime characteristicEdens, Oakley0Reichstein, Zinovy1https://orcid.org/0000-0002-3157-2066Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, CanadaDepartment of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, CanadaThe essential dimension $\operatorname{ed}_k(\operatorname{S}_n)$ of the symmetric group $\operatorname{S}_n$ is the minimal integer $d$ such that the general polynomial $x^n + a_1 x^{n-1} + \dots + a_n$ can be reduced to a $d$-parameter form by a Tschirnhaus transformation. Finding this number is a long-standing open problem, originating in the work of Felix Klein, long before essential dimension of a finite group was formally defined. We now know that $\operatorname{ed}_k(\operatorname{S}_n)$ lies between $\lfloor n/2 \rfloor $ and $n-3$ for each $n \geqslant 5$ and any field $k$ of characteristic different from $2$. Moreover, if $\operatorname{char}(k) = 0$, then $\operatorname{ed}_k(\operatorname{S}_n) \geqslant \lfloor (n+1)/2 \rfloor $ for any $n \geqslant 7$. The value of $\operatorname{ed}_k(\operatorname{S}_n)$ is not known for any $n \geqslant 8$ and any field $k$, though it is widely believed that $\operatorname{ed}_k(\operatorname{S}_n)$ should be $n-3$ for every $n \geqslant 5$, at least in characteristic $0$. In this paper we show that for every prime $p$ there are infinitely many positive integers $n$ such that $\operatorname{ed}_{\mathbb{F}_p}(\operatorname{S}_n) \leqslant n-4$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.577/Essential dimensionsymmetric groupgeneral polynomialgroup action on an algebraic varietypositive characteristic |
| spellingShingle | Edens, Oakley Reichstein, Zinovy Essential dimension of symmetric groups in prime characteristic Comptes Rendus. Mathématique Essential dimension symmetric group general polynomial group action on an algebraic variety positive characteristic |
| title | Essential dimension of symmetric groups in prime characteristic |
| title_full | Essential dimension of symmetric groups in prime characteristic |
| title_fullStr | Essential dimension of symmetric groups in prime characteristic |
| title_full_unstemmed | Essential dimension of symmetric groups in prime characteristic |
| title_short | Essential dimension of symmetric groups in prime characteristic |
| title_sort | essential dimension of symmetric groups in prime characteristic |
| topic | Essential dimension symmetric group general polynomial group action on an algebraic variety positive characteristic |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.577/ |
| work_keys_str_mv | AT edensoakley essentialdimensionofsymmetricgroupsinprimecharacteristic AT reichsteinzinovy essentialdimensionofsymmetricgroupsinprimecharacteristic |