Essential dimension of double covers of symmetric and alternating groups

TL;DR

This paper investigates the essential dimension of double covers of symmetric and alternating groups, revealing exponential growth in characteristic not 2 and sublinear growth in characteristic 2, with applications to trace forms.

Contribution

It provides the first detailed analysis of the essential dimension of these double covers, highlighting their growth behavior across different characteristics.

Findings

Exponential growth of essential dimension in characteristic not 2

Sublinear growth of essential dimension in characteristic 2

Application to trace form theory in good characteristic

Abstract

I. Schur studied double covers $\widetilde{\Sym}^{\pm}_n$ and $\widetilde{\Alt}_n$ of symmetric groups $\Sym_n$ and alternating groups $\Alt_n$, respectively. Representations of these groups are closely related to projective representations of $\Sym_n$ and $\Alt_n$; there is also a close relationship between these groups and spinor groups. We study the essential dimension $\ed(\widetilde{\Sym}^{\pm}_n)$ and $\ed(\widetilde{\Alt}_n)$. We show that over a base field of characteristic $\neq 2$, $\ed(\widetilde{\Sym}^{\pm}_n)$ and $\ed(\widetilde{\Alt}_n)$ grow exponentially with $n$, similar to $\ed(\Spin_n)$. On the other case, in characteristic $2$, they grow sublinearly, similar to $\ed(\Sym_n)$ and $\ed(\Alt_n)$. We give an application of our result in good characteristic to the theory of trace forms.