## 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 where takes the value one if and are conjugate and zero otherwise and is the centralizer of in .

*Motivation*

In the past we’ve seen that if we ‘fix and let ‘vary’ over (in the form of the sum) that there is an interesting orthogonality relation. Namely, this is just the orthogonality relation . In this post we explore what happens if the ‘fixing’ and ‘varying’ are reversed. In other words, we compute . 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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