Essential dimension of symmetric groups in prime characteristic

Abstract

The essential dimension ${\mathrm{ed}}_k({\mathrm{S}}_n)$ of the symmetric group ${\mathrm{S}}_n$ 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 was formally defined. We now know that ${\mathrm{ed}}_k({\mathrm{S}}_n)$ lies between $\lfloor n/2\rfloor$ and $n-3$ for every $n⩾5$ and every field $k$ of characteristic different from $2$. Moreover, if $\mathrm{char}(k)=0$, then ${\mathrm{ed}}_k({\mathrm{S}}_n)⩾\lfloor (n+1)/2\rfloor$ for any $n⩾6$. The value of ${\mathrm{ed}}_k({\mathrm{S}}_n)$ is not known for any $n⩾8$ and any field $k$, though it is widely believed that ${\mathrm{ed}}_k({\mathrm{S}}_n)$ should be $n-3$ for every $n⩾5$, at least in characteristic $0$. In this paper we show that for every odd prime $p$ there are infinitely many positive integers $n$ such that ${\mathrm{ed}}_{{\mathbb{F}}_p}({\mathrm{S}}_{\mathrm{n}})⩽\mathrm{n}-4$.

8 pages