Thursday April 27 - no class next Tuesday

Side note: to compute the character of a GL(n) rep on an arbitrary element of GL(n), take its eigenvalues and feed that diagonal matrix into the WCF.

Given the multiplicity diagram of a G-rep V, apply difference operators in the direction of the negative roots, and the result (in the dominant Weyl chamber) tells you the multiplicities of the irreps in V.
The branching rule from GL(n) to GL(n-1).
The weight of a Gel'fand-Cetlin pattern is the list of differences in the row sums.
Theorem: there is a basis of V_lambda indexed by Gel'fand-Cetlin patterns, where each basis vector is a T-weight vector whose weight is that of the GC pattern.

I am away next Tuesday.