Thursday April 13

Gah! I'm already slipping on keeping this blog up-to-date.

Lemma (b). The "fundamental representations" Alt^k Cⁿ have weights all of multiplicity 1. The high weight is (1,...,1,0,...,0) with k 1s and n-k 0s. The other weights are permutations thereof, and all other weights are less than the high weight in dominance order.

Lemma (c). Call a rep "good" if the high weight is multiplicity one, and all other weights are smaller in dominance order. (So the high weight is unique.) Then the tensor product of two such reps is again "good", and the new high weight is the sum of the two individual high weights.

Theorem 1. Every irrep has a dominant high weight.

Theorem 2. For every dominant weight, there is an irrep with that high weight, and it's "good".