Totally symmetric sets in the general linear group

TL;DR

This paper studies totally symmetric sets within the general linear group, introducing a broader perspective and classifying irreducible and maximal cases, revealing their algebraic rigidity and structural properties.

Contribution

It introduces a generalized concept of total symmetry and classifies irreducible totally symmetric sets in the general linear group.

Findings

Classified irreducible totally symmetric sets.

Identified maximal cardinality of such sets.

Showed rigidity under group homomorphisms.

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.