Definition:Dimension (Representation Theory)

Definition

Let $(k,+,\circ)$ be a field.

Let $V$ be a vector space over $k$ of finite dimension.

Let $\operatorname {GL} \left({V}\right)$ be the general linear group of $V$.

Let $(G, \cdot)$ be a finite group.

Let $\rho : G \to \operatorname {GL} \left({V}\right)$ be a linear representation of $G$ on $V$.

The dimension or degree of $\rho$, written $\deg(\rho)$ is the dimension of the vector space $V$.