On base sizes for algebraic groups

Timothy C. Burness, Robert M. Guralnick, Jan Saxl

Research output: Contribution to journalArticle (Academic Journal)

11 Citations (Scopus)
183 Downloads (Pure)

Abstract

Let G be a permutation group on a set Ω. A subset of Ω is a base for G if its pointwise stabilizer is trivial; the base size of G is the minimal cardinality of a base. In this paper we initiate the study of bases for algebraic groups defined over an algebraically closed field. In particular, we calculate the base size for all primitive actions of simple algebraic groups, obtaining the precise value in almost all cases. We also introduce and study two new base measures, which arise naturally in this setting. We give an application concerning the essential dimension of simple algebraic groups, and we establish several new results on base sizes for the corresponding finite groups of Lie type. The latter results are an important contribution to the classical study of bases for finite primitive permutation groups. We also indicate some connections with generic stabilizers for representations of simple algebraic groups.

Original languageEnglish
Pages (from-to)2269-2341
Number of pages73
JournalJournal of the European Mathematical Society
Volume19
Issue number8
DOIs
Publication statusPublished - 1 Aug 2017

Keywords

  • Base size
  • Generic stabilizer
  • Primitive permutation groups
  • Simple algebraic groups

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