Degree of a linear representation

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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