Representative function

Definition
Let $$G$$ be a group, $$k$$ a field, $$\rho$$ a representation of $$G$$ over $$k$$ such that every element of $$G$$ preserves an inner product over $$k$$. A representative function for $$\rho$$ is defined as a function that can be described as one of the matrix entry functions (relative to an orthonormal basis) corresponding to $$\rho$$.

Equivalently, a representative function is a function defined as the inner product of $$\rho(e)$$ and $$f$$ where $$e$$ and $$f$$ are either orthogonal or equal unit vectors.

The dimension of the space of representative functions equals the square of the degree of the representation. More specifically, we claim that the matrix entries for any particular choice of orthonormal basis form a basis for the space of representative functions for the representations.