Exercises #3

1. Show that dominance order is indeed a partial order.

2. Let lambda be dominant, and mu a convex combination of the permutations of lambda.
(This means: let x_pi be a nonnegative real number for each permutation pi in S_n, with \sum_pi x_pi = 1. Let mu = \sum_pi x_pi pi.lambda . Then mu is a "convex combination".)
Show that lambda is more than mu in dominance order.

3. Call a weight multiplicity diagram D "saturated" if the conditions
a) mu is integral
b) mu is a convex combination of weights in D
together imply that mu is itself a weight of D.
Show that Symⁿ C² is the smallest S₂-symmetric saturated multiplicity diagram with high weight (n,0), for some appropriate definition of "smallest".

4. Show that the reducible representations we made by tensoring fundamental representations together are saturated.