Tuesday April 25

Statement of the Weyl character formula (WCF) with a single denominator, Den.
The Kostant partition function.
The Kostant multiplicity formula, as the Fourier transform of WCF.
Pictures of the weight multiplicity diagrams as given by the Kostant multiplicity formula, for the GL(3) case.
Proof that the WCF is actually a Laurent polynomial with integer coefficients (given by the Kostant multiplicity formula).
Weyl's proof that the WCF is the character of an irrep, indeed of V_lambda.

Gel'fand-Cetlin patterns, the integer points in the (compact, convex) Gel'fand-Cetlin polytope.
Example: Gel'fand-Cetlin polytopes for GL(3).