Abstract Nonsense

Crushing one theorem at a time

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.

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March 7, 2011 Posted by | Algebra, Representation Theory | , , , , | 5 Comments