Representation over the rational numbers ($\rho :G\to \mathrm{GL}(V_{\mathbb{Q}})$)

A representation over the rational numbers is a group representation on a finite-dimensional rational vector space. Choosing a basis gives rational matrices. It differs from a rational representation of an algebraic group, where rational refers to regular matrix-coefficient functions rather than the ground field. Scalar extension yields representations over every characteristic-zero field.