Hostname: page-component-7c8c6479df-5xszh Total loading time: 0 Render date: 2024-03-29T12:18:59.226Z Has data issue: false hasContentIssue false

Zero-Separating Invariants for Linear Algebraic Groups

Published online by Cambridge University Press:  22 December 2015

Jonathan Elmer
Affiliation:
University of Aberdeen, King's College, Aberdeen AB24 3UE, UK (j.elmer@abdn.ac.uk)
Martin Kohls
Affiliation:
, Technische Universität München, Zentrum Mathematik-M11, Boltzmannstrasse 3, 85748 Garching, Germany (kohls@ma.tum.de)

Abstract

Abstract Let G be a linear algebraic group over an algebraically closed field 𝕜 acting rationally on a G-module V with its null-cone. Let δ(G, V) and σ(G, V) denote the minimal number d such that for every and , respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) ≠ 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V. For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL2(𝕜) that contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G0 is unipotent. Our results also lead to a more elementary proof that βsep(G) is finite if and only if G is finite.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)