The character ${\chi }_V(g)=\mathrm{tr}({\rho }_V(g))$ of a finite-dimensional complex representation

Let $\rho :G\to \mathrm{GL}(V)$ be a finite-dimensional complex representation (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree). Its character is the function

${\chi }_V:G⟶\mathbb{C},{\chi }_V(g):=\mathrm{tr}\text{\hspace{0.17em}⁣}({\rho }_V(g)),$

where the trace is the basis-independent trace of an endomorphism (The basis-independent trace of an endomorphism of a finite-dimensional vector space); one writes ${\chi }_{\rho }$ or $\chi$ when the representation or the group is fixed.

The definition is well posed in two senses, recorded here because both are used throughout the page. First, the value does not depend on a choice of basis of $V$: matrices of one endomorphism in two ordered bases are similar, and similar matrices have equal trace (The basis-independent trace of an endomorphism of a finite-dimensional vector space). Second, equivalent representations have equal characters: if $T:V\to W$ is an invertible intertwiner, then ${\rho }_W(g)=T{\rho }_V(g)T^{-1}$, so $\mathrm{tr}({\rho }_W(g))=\mathrm{tr}({\rho }_V(g)T^{-1}T)=\mathrm{tr}({\rho }_V(g))$ by the identity $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ (For $A\in M_{m\times n}(F)$ and $B\in M_{n\times m}(F)$, $\mathrm{tr}(AB)=\mathrm{tr}(BA)$). Thus the character depends only on the equivalence class of the representation.