Essential dimension of symmetric groups in prime characteristic

Oakley Edens; Zinovy Reichstein

doi:10.5802/crmath.577

Research article - Algebra

Essential dimension of symmetric groups in prime characteristic

Oakley Edens ¹; Zinovy Reichstein ¹

¹ Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 639-647.

Abstract
Résumé

The essential dimension ${ed}_{k} (S_{n})$ of the symmetric group $S_{n}$ 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 ${ed}_{k} (S_{n})$ lies between $⌊ n / 2 ⌋$ and $n - 3$ for each $n ⩾ 5$ and any field $k$ of characteristic different from $2$ . Moreover, if $char (k) = 0$ , then ${ed}_{k} (S_{n}) ⩾ ⌊ (n + 1) / 2 ⌋$ for any $n ⩾ 7$ . The value of ${ed}_{k} (S_{n})$ is not known for any $n ⩾ 8$ and any field $k$ , though it is widely believed that ${ed}_{k} (S_{n})$ should be $n - 3$ for every $n ⩾ 5$ , at least in characteristic $0$ . In this paper we show that for every prime $p$ there are infinitely many positive integers $n$ such that ${ed}_{𝔽_{p}} (S_{n}) ⩽ n - 4$ .

La dimension essentielle ${ed}_{k} (S_{n})$ du groupe symétrique $S_{n}$ est le plus petit entier $d$ permettant de réduire le polynôme général $x^{n} + a_{1} x^{n - 1} + \dots + a_{n}$ à une forme comportant $d$ paramètres par une transformation de Tschirnhaus. La détermination de cette valeur est un problème ouvert depuis longtemps, remontant aux recherches de Felix Klein, bien avant que la dimension essentielle d’un groupe fini ne soit formellement définie. Nous savons que ${ed}_{k} (S_{n})$ se situe entre $⌊ n / 2 ⌋$ et $n - 3$ pour tout entier $n ⩾ 5$ et tout corps $k$ de caractéristique autre que $2$ . De plus, si $char (k) = 0$ , on sait que ${ed}_{k} (S_{n}) ⩾ ⌊ (n + 1) / 2 ⌋$ pour tout $n ⩾ 7$ . La valeur de ${ed}_{k} (S_{n})$ est inconnue dès que $n ⩾ 8$ ceci quelque soit le corps $k$ ; bien qu’on estime généralement que ${ed}_{k} (S_{n})$ devrait être $n - 3$ pour tout $n ⩾ 5$ , au moins en caractéristique $0$ . Nous démontrons que pour tout nombre premier $p$ , il existe une infinité d’entiers positifs $n$ tels que ${ed}_{𝔽_{p}} (S_{n}) ⩽ n - 4$ .

Received: 2023-08-28
Accepted: 2023-10-09
Revised after acceptance: 2023-10-31
Published online: 2024-07-09

DOI: 10.5802/crmath.577

Classification: 12E05, 14G17, 14L30, 14E05
Keywords: Essential dimension, symmetric group, general polynomial, group action on an algebraic variety, positive characteristic
Mots-clés : Dimension essentielle, groupe symétrique, polynôme général, action d’un groupe sur une variété algébrique, caractéristique positive

Author's affiliations:

Oakley Edens ¹; Zinovy Reichstein ¹

¹ Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Oakley Edens and Zinovy Reichstein},
     title = {Essential dimension of symmetric groups in prime characteristic},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {639--647},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2024},
     doi = {10.5802/crmath.577},
     language = {en},
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Oakley Edens; Zinovy Reichstein. Essential dimension of symmetric groups in prime characteristic. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 639-647. doi : 10.5802/crmath.577. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.577/

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