A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree

Let $k$ be a field (Field) and let $G$ be a group. A finite-dimensional representation of $G$ over $k$ is a finite-dimensional $k$-vector space $V$ (Vector space over a field, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis) together with a group homomorphism $\rho :G⟶\mathrm{GL}(V),$ where $\mathrm{GL}(V)$ denotes the group of invertible $k$-linear maps $V\to V$ (Invertible linear maps, linear isomorphisms, and inverse linear maps, Monoid homomorphism and group homomorphism).

The associated action is $g\cdot v:=\rho (g)(v)\mathrm{.}$ This makes $V$ into a $G$-module over $k$ in the sense of An $R$-linear action of $G$ on a left $R$-module, and a $G$-module over $R$.

The degree of the representation is the dimension of its underlying vector space: $\mathrm{deg}(\rho ):={\dim }_{k}V\mathrm{.}$

• The finite-dimensionality convention is part of the definition on this page, not a later standing assumption.

• The representation is determined equally well by the homomorphism $\rho$ or by the associated $k$-linear action of $G$ on $V$.