In an attempt to understand the plethysm coefficients s_w[s_h] we began by studying the restriction of this character to SL_2. Those characters end up being q-binomial coefficients which are the generating functions for partitions in a wxh rectangle. The most beautiful structure we could hope for would be a symmetric chain decomposition on the poset of partitions in a wxh rectangle. Strangely, this problem ends up being very hard and, while it is known that the poset is Sperner, it is not known if there exists a symmetric chain decomposition in general. I will explain some recent conjectures about q-binomial coefficients (which we can prove in small cases) and how it might help resolve this conjecture.

This is joint work with Álvaro Gutiérrez, Franco Saliola, Anne Schilling and Rosa C. Orellana