Representation theory: Orthogonality relations
Channel: Richard E Borcherds
Date: 2020-09-09

This lecture is about the orthogonality relations of the character table of complex representations of a finite group. 
We show that these representations are unitary and deduce that they are all sums of irreducible representations. We then prove Schur's lemma describing the dimension of the space of invariant maps between irreducible representations, and deduce that the rows of the character table are orthogonal. We show that the regular representation decomposes as a sum of all irreducible representations with multiplicity given by the dimension, and by examining the center of the group algebra show that the number of irreducible representations is the number of conjugacy classes. From this we deduce the orthogonality of the rows. 

As an example, we show how to find the dimensions of irreducible representations of the alternating group A5 from just knowing its order and the number of conjugacy classes.