Invariants of wreath products and subgroups of S_6

TL;DR

This paper studies the rationality of fixed fields under subgroup actions of S_6, identifying exceptions and exploring wreath product invariants, with implications for algebraic geometry and invariant theory.

Contribution

It classifies when the fixed field of a subgroup of S_6 acting on rational functions is rational or stably rational, and investigates invariants of wreath products.

Findings

Fixed fields are rational except for specific subgroups isomorphic to PSL_2(F_5), PGL_2(F_5), or A_6.

For PSL_2(F_5) and PGL_2(F_5), fixed fields are stably rational over any field.

The invariant theory of wreath products is also analyzed.

Abstract

Let $G$ be a subgroup of $S_6$, the symmetric group of degree 6. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,...,x_6)$ via $k$-automorphisms defined by $\sigma\cdot x_i=x_{\sigma(i)}$ for any $\sigma\in G$, any $1\le i\le 6$. Theorem. The fixed field $k(x_1,...,x_6)^G$ is rational (=purely transcendental) over $k$, except possibly when $G$ is isomorphic to $PSL_2(\bm{F}_5)$, $PGL_2(\bm{F}_5)$ or $A_6$. When $G$ is isomorphic to $PSL_2(\bm{F}_5)$ or $PGL_2(\bm{F}_5)$, then $\bm{C}(x_1,...,x_6)^G$ is $\bm{C}$-rational and $k(x_1,...,x_6)^G$ is stably $k$-rational for any field $k$. The invariant theory of wreath products will be investigated also.