Representation theory: The Schur indicator
Channel: Richard E Borcherds
Date: 2020-09-12

This is about the Schur indicator of a complex representation. 
It can be used to check whether an irreducible representation has in invariant bilinear form, and if so whether the form is symmetric or antisymmetric. 
As examples we check which representations of the dihedral group D8, the quaternion group Q8, and the alternating group A4 have symmetric or antisymmetric bilinear forms. 

We discuss representations of odd groups, showing that all nontrivial irreducible representations have Schur indicator 0, and use thi to show that the number of conjugacy classes is equal to the number of elements mod 16.

Finally we describe how to use the Schur indicator to find the real irreducibel representations of a group, and do the cases of Q8 and A4 as examples.