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THE ${\it\alpha}$ -INVARIANT AND THOMPSON’S CONJECTURE

Part of: General commutative ring theory Representation theory of groups Local theory

Published online by Cambridge University Press:  08 July 2016

PHAM HUU TIEP
PHAM HUU TIEP*
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA; tiep@math.arizona.edu

Abstract

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In 1981, Thompson proved that, if $n\geqslant 1$ is any integer and $G$ is any finite subgroup of $\text{GL}_{n}(\mathbb{C})$ , then $G$ has a semi-invariant of degree at most $4n^{2}$ . He conjectured that, in fact, there is a universal constant $C$ such that for any $n\in \mathbb{N}$ and any finite subgroup $G<\text{GL}_{n}(\mathbb{C})$ , $G$ has a semi-invariant of degree at most $Cn$ . This conjecture would imply that the ${\it\alpha}$ -invariant ${\it\alpha}_{G}(\mathbb{P}^{n-1})$ , as introduced by Tian in 1987, is at most $C$ . We prove Thompson’s conjecture in this paper.

MSC classification

Primary: 20C15: Ordinary representations and characters
Secondary: 13A50: Actions of groups on commutative rings; invariant theory 14B05: Singularities 20C33: Representations of finite groups of Lie type
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 4 , 2016 , e5
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2016

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