Feb 10

Thm. Multiplicity diagrams of U(n) reps are S_n-symmetric.
Proof. Use S_n to define maps between the weight spaces.

Thm.
1. If V is pointed at lambda, then V contains a unique irrep pointed at lambda.
2. If V,W are pointed at lambda,mu, then V tensor W is pointed at lambda+mu.
3. Alt^k(C^n) is pointed at (1,...1,0,...0) with k 1s.

Thm.
For every weakly decreasing lambda, there exists a unique irrep pointed at lambda, and these are all the irreps of U(n) or GL(n).

Proof. Tensor together Alt^ks to produce a rep pointed at lambda.
Let V be any rep; we'll try to write it as a sum of these V_lambda.
It's enough to check U(n)-isomorphism, and for that, it's enough to check the character, i.e. the multiplicity diagram.
Prove inductively that we can write any S_n-symmetric Z-valued function on Z^n, of finite support, as a linear combination of characters of these V_lambda.
Since irreducible characters are orthogonal, there can't be any other irreps.