Exercises #5

1. Use the Steinberg rule to prove the Pieri rule.
2. Show that the hive theorem is true for the following cases:

tensoring with the trivial representation

the case V_(2,1,0)^@2

the Pieri cases.

3. A hive is defined by its convexity. Hence, if we add a linear function to a hive we get another one. What is the corresponding statement about tensor products?
4. Fix lambda,mu,nu. Show that the space of real hives with this boundary is bounded (hence compact).
5*. Fix lambda,mu,kappa. What does the Steinberg rule say about the limit of
c_{N lambda,mu}^{N lambda + kappa} as N -> infinity? (We dealt with the case that lambda is regular dominant, i.e. has no repeats.)