## 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.

*Second Orthogonality Relation for the Irreducible Characters*

**Theorem (Second Orthogonality Relation for the Irreducible Characters): ***For a finite group and denote the centralizer of by . Then, for let take the value one if and are conjugate and zero otherwise. Then, *

**Proof: **Let be the number of conjugacy classes in and let be a set of representatives such that for . Recall then that and so the elements of can be labeled . We then consider the matrix where . We then let denote the matrix . Recall though from the first orthogonality relation that

But, recalling that the irreducible characters are class functions and thus constant on conjugacy classes it easily follows that this may be rewritten

This then implies that

But, recall the fact that for square matrices a left inverse is necessarily a right inverse, and thus the above implies that

But, the general entry of this right hand matrix is

Dividing both sides and recalling by the orbit stabilizer theorem that we see that

And recalling all of our notation and taking the conjugate of both sides gives the desired result.

**References:**

1. Isaacs, I. Martin. *Character Theory of Finite Groups*. New York: Academic, 1976. Print.

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