# Equivalence of linear representations

## Contents

## Definition

### In terms of a homomorphism of linear representations

An **equivalence of linear representations** between a linear representation and a linear representation (where is a group and are vector spaces over a field ) is a homomorphism of linear representations from to having a two-sided inverse that is also a homomorphism of linear representations. In other words, it is a vector space isomorphism such that:

### For representations given as matrices

Given two linear representations and of a group over a field , an equivalence of representations between is given by a matrix such that:

### Equivalence of definitions

The second definition is a special case of the first if we view and as the matrix for the isomorphism .

### Notion of equivalent linear representations

Two linear representations are said to be *equivalent* if there exists an equivalence of linear representations.

Most notions related to linear representations are studied up to equivalence. When we say that two representations are distinct, or seek to count the *number* of representations of a certain type, we are doing this up to equivalence.

## Facts

- Character determines representation in characteristic zero
- Equivalent linear representations of finite group over field are equivalent over subfield in characteristic zero: In particular, at least in characteristic zero, the notion of being equivalent does not depend on which field we are thinking of our matrices as living in.