Zero-Separating Invariants for Linear Algebraic Groups

Jonathan Elmer, Martin Kohls

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Let G be a linear algebraic group over an algebraically closed field 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) = â for any subgroup G of GL2() 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 G is unipotent. Our results also lead to a more elementary proof that β sep(G) is finite if and only if G is finite.

Original languageEnglish
Pages (from-to)911-924
Number of pages14
JournalProceedings of the Edinburgh Mathematical Society
Volume59
Issue number4
Early online date22 Dec 2015
DOIs
Publication statusPublished - 1 Nov 2016

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Linear Algebraic Groups
Invariant
Zero
If and only if
Denote
Module
Infinite Groups
Positive Characteristic
Supremum
Algebraically closed
Null
Cone
Subgroup

Keywords

  • geometrically reductive
  • global degree bounds
  • invariant theory
  • linear algebraic groups
  • prime characteristic

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Zero-Separating Invariants for Linear Algebraic Groups. / Elmer, Jonathan; Kohls, Martin.

In: Proceedings of the Edinburgh Mathematical Society, Vol. 59, No. 4, 01.11.2016, p. 911-924.

Research output: Contribution to journalArticle

Elmer, Jonathan ; Kohls, Martin. / Zero-Separating Invariants for Linear Algebraic Groups. In: Proceedings of the Edinburgh Mathematical Society. 2016 ; Vol. 59, No. 4. pp. 911-924.
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