How to compute this character?

Question

Let $K$ a field of characteristic $0$ and $K[G]$ a group algebra of a finite group $G$. Let $I$ be a left ideal of $K[G]$ generated by an idempotent $e$ and $\rho: G \longrightarrow GL(I)$ a representation defined by $\rho(g)(ae)=gae$. Does someone know how to compute $\chi_{\rho}$ ? Many thanks in advance !

Answer 1

For each $g \in G$, the value $\chi(g)$ is the trace of the composed map $$ I \to K[G] \stackrel{g\cdot }{\to} K[G] \stackrel{\cdot e}{\to} I, $$ where the first map is the inclusion map. By the usual trace properties, it can also be computed as the trace of the composed map $$ K[G] \stackrel{\cdot e}{\to} I \to K[G] \stackrel{g\cdot }{\to} K[G]. $$ Now we may use the standard basis $G$ of $K[G]$. For each $h \in G$ and $x \in K[G]$ let $\langle x , h \rangle \in K$ denote the coefficient of $h$ in $x$ (which of course may also be represented by a formula such as $\langle x , h \rangle = \rho(xh^{-1}) / |G|$, where $\rho$ is the regular character of $G$). With that notation, we have $$ \chi(g) = \sum_{h \in G} \langle ghe, h \rangle. $$