Character of a representation

Last updated January 27, 2022

Idea

Eigenvalues play a very important role in understanding invariant subspaces of a representation, and thus the representations themselves. Calculating eigenvalues for every element of $G$ can be very tedious, though. The key observation is that it suffices to know the sum of eigenvalues for every element in $g$.

Definition

Let $T$ be a representation of $G$ on $V$.

Definition. The character of $T$ is the function \begin{gather}\chi_V:G\to\mathbb{F} \ g\mapsto \text{Tr}(Tg).\end{gather}

Properties

1. $\chi_{V\oplus W}=\chi_V+\chi_W$
2. $\chi_{V\otimes W}=\chi_V\cdot \chi_W$
3. $\chi_{V^\ast}=\overline{\chi}_V$
4. $\chi_{\Lambda^2V}(g)=\frac{1}{2}[\chi_V(g)^2]-\chi_V(g^2)]$
5. If $G$ is a finite group and $T$ is a complex representation, then there exists an inner product in $V$ such that $T(g)$ is unitary for every $g\in G$.