Feb 15, 17, 21

Gel'fand-Cetlin patterns.
Theorem: they compute weight multiplicities iff V_lambda (for GL(n)) decomposes as the sum of V_mu (for GL(n-1)) where mu intersperses between lambda.
(We'll prove this later.)

Weyl integration formula.

Bruhat decomposition for GL(n).

The character of T acting on a polynomial ring is formally a rational function.

The weight multiplicities in V_lambda are bounded above by those in a certain polynomial ring.

That's not S_n invariant; so let's hit it with S_n to symmetrize it.
Theorem next time: the Weyl character formula, which says that that's the right answer.