Class-determining field

This term is related to: linear representation theory
View other terms related to linear representation theory | View facts related to linear representation theory

This term associates to every group, a corresponding field property. In other words, given a field, every field either has the property with respect to that group or does not have the property with respect to that group

Definition

Symbol-free definition

A field is said to be a class-determining field for a group if any representation is determined by the conjugacy classes in which the images of the conjugacy classes of the group lie under that representation.

In other words, for any two distinct (i.e. inequivalent) linear representations, there exists a conjugacy class whose image under the two representations does not lie in the same conjugacy class in the general linear group.

Equivalently, no two inequivalent linear representations are locally conjugate.

Definition with symbols

A field $k$ is termed a class-determining field for a group $G$ if for any two finite-dimensional linear representations ${\rho }_1,{\rho }_2:G\to GL(V)$, there exists $g\in G$ such that ${\rho }_1(g)$ and ${\rho }_2(g)$ are not conjugate.

Relation with other properties

Stronger properties

• Character-determining field

Facts

For a finite group, any field of coprime characteristic is character-determining, and hence also class-determining.