Twisted symmetric group actions

Akinari Hoshi and Ming-chang Kang

Vol. 248 (2010), No. 2, 285–304

DOI: 10.2140/pjm.2010.248.285

Abstract

Let K be any field, let K(x₁,…,x_n) be the rational function field of n variables over K, and let S_n and A_n be the symmetric group and the alternating group of degree n, respectively. For any a ∈ K ∖{0}, define an action of S_n on K(x₁,…,x_n) by σ ⋅ x_i = x_σ(i) for σ ∈ A_n and σ ⋅ x_i = a∕x_σ(i) for σ ∈ S_n ∖ A_n. We prove that for any field K and n = 3,4,5, the fixed field K(x₁,…,x_n)^S_n is rational (that is, purely transcendental) over K.

Keywords

rationality problem, conic bundles

Mathematical Subject Classification 2000

Primary: 13A50, 12F20

Milestones

Received: 2 September 2009

Revised: 22 December 2009

Accepted: 26 December 2009

Published: 1 December 2010