Let $\rho:G\longrightarrow\operatorname{GL}(V)$ be a finite dimensional representation of a group $G$ (i.e., $V$ is a finite dimensional vector space over its scalar field $K$). The character of $\rho$ is the function $\chi_{V}:G\longrightarrow K$ defined by

where $\operatorname{Tr}$ is the trace function.

Properties:

• $\chi_{V}(g)=\chi_{V}(h)$ if $g$ is conjugate to $h$ in $G$. (Equivalently, a character is a class function on $G$.)

• If $G$ is finite, the characters of the irreducible representations of $G$ over the complex numbers form a basis of the vector space of all class functions on $G$ (with pointwise addition and scalar multiplication).

• Over the complex numbers, the characters of the irreducible representations of $G$ are orthonormal under the inner product

  $$(\chi_{1},\chi_{2}):=\frac{1}{|G|}\sum_{{g\in G}}\overline{\chi_{1}(g)}\chi_{2}(g)$$