On base sizes for algebraic groups
Timothy C. Burness
University of Bristol, UKRobert M. Guralnick
University of Southern California, Los Angeles, USAJan Saxl
University of Cambridge, UK
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Abstract
Let be a permutation group on a set . A subset of is a base for if its point-wise stabilizer is trivial; the base size of 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.
Cite this article
Timothy C. Burness, Robert M. Guralnick, Jan Saxl, On base sizes for algebraic groups. J. Eur. Math. Soc. 19 (2017), no. 8, pp. 2269–2341
DOI 10.4171/JEMS/718