character

Let $\rho :G⟶\mathrm{GL}(V)$ 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 ${\chi }_V:G⟶K$ defined by

$${\chi }_V(g):=\mathrm{Tr}(\rho (g))$$

where $\mathrm{Tr}$ 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

  $$({\chi }_1,{\chi }_2):=\frac{1}{|G|}\sum _{g\in G}\overline{{\chi }_1(g)}{\chi }_2(g)$$