# Locally conjugate representations

This term is related to: linear representation theory

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## Definition

### Symbol-free definition

Two linear representations of a group over a field are said to be **locally conjugate** if for any element of the group, its image under the two representations defines equivalent linear transformations (in particular, if we identify the vector spaces on which the representations are there, its images must be conjugate).

### Definition with symbols

Let be a group and a field.

Two linear representations and of over are locally conjugate if for any , there exists such that:

### Definition using the L-map

Let be a group and a field. Let denote the set of conjugacy classes of and the set of indecomposable linear representations of over .

Let denote the conjugacy class of in the general linear group for any (they are all in the same conjugacy class). is thus a map:

where is the union of the set of conjugacy classes in for all vector spaces over .

Then, we say that two representations and are **locally conjugate** with respect to if for any :

Note that though the above is described in terms of indecomposable linear representation, we can define the -map for arbitrary linear representations, and the same definition can thus be used to call two arbitrary linear representations locally conjugate.