Thursday April 20

Lemma. Let D be an S_n-invarant weight multiplicity diagram with some D^mu > 0. Then D defines a class function Chi_D:U(n) -> C. If the integral over U(n) of |Chi_D|² = 1, then D is the weight diagram of an irrep.
Weyl integration formula, in terms of volumes of conjugacy classes.
Volumes of conjugacy classes of U(n). (statement, no proof; look in the notes on the class web page for full details)
The Weyl denominator Den, as a sort of square root of the volume of the conjugacy class.
Statement of the Weyl character formula (WCF), manifestly S_n-invariant version.
Theorem. GL(2) acting on Sym^k C² is an irrep.
The WCF for GL(2).
The rho-shifting in Den * WCF.