# Degree of a linear representation

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## Contents

This article gives a basic definition in the following area: linear representation theory
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## Definition

### Symbol-free definition

The degree of a linear representation is defined as the dimension of the vector space to which the representation's map is defined.

It also equals the value of the character at the identity element.

### Definition with symbols

Suppose $(V,\rho)$ is a linear representation of a group $G$ over a field $k$, i.e. we have a homomorphism $\rho:G \to GL(V)$, where $V$ is a vector space over $k$. Then, the degree of the representation $(V,\rho)$ is defined as the dimension of $V$ as a $k$-vector space.

For a finite group, the degrees of irreducible representations are important numbers. Further information: degrees of irreducible representations