# Abstract Nonsense

## Second Orthogonality Relation For Irreducible Characters

Point of post: In this post we discuss the second orthogonality relation for the irreducible characters. In particular, we prove that $\displaystyle \sum_{\alpha\in\widehat{G}}\chi^{(\alpha)}(g)\overline{\chi^{(\alpha)}(h)}=c(g,h)\#\left(\bold{C}_G(g)\right)$ where $c(g,h)$ takes the value one if $g$ and $h$ are conjugate and zero otherwise and $\bold{C}_G(g)$ is the centralizer of $g$ in $G$.

Motivation

In the past we’ve seen that if we ‘fix $\alpha,\beta\in\widehat{G}$ and let $g$ ‘vary’ over $G$ (in the form of the sum) that there is an interesting orthogonality relation. Namely, this is just the orthogonality relation $\displaystyle \frac{1}{|G|}\sum_{g\in G}\chi^{(\alpha)}(g)\overline{\chi^{(\beta)}(g)}=\delta_{\alpha,\beta}$. In this post we explore what happens if the ‘fixing’ and ‘varying’ are reversed. In other words, we compute $\displaystyle \sum_{\alpha\in\widehat{G}}\chi^{(\alpha)}(g)\overline{\chi^{(\alpha)}(h)}$. This shall serve as an interesting tool inall that comes. We call this the second orthogonality relation for the irreducible characters.