Totally symmetric sets in the general linear group
arXiv:2202.10216
Abstract
A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group. Such sets are rigid under homomorphisms, and so exert a great deal of control over the algebraic structure. In this paper we introduce a more general perspective on total symmetry, and formulate a notion of "irreducibility" for totally symmetric sets in the general linear group. We classify irreducible totally symmetric sets, as well as those of maximal cardinality.
40 pages plus 6 page appendix, no figures. New in v2: gap repaired - one new example in Theorem B, results otherwise unaffected. Second half of the paper restructured and rewritten. Comments welcome!