This article is based on my talk at the workshop on Representation Theory and Symmetric Functions, MSRI, April 14, 1997. I thank the organizers (Sergey Fomin, Curtis Greene, Phil Hanlon and Sheila Sundaram) for bringing together a group of outstanding combinatorialists and for giving me a chance to bring to their attention some of the problems that I find very exciting and beautiful. In preparing the note for this volume (October 1998), I made a few small changes in the original version [Zelevinsky 1997], and added in the end a brief (and undoubtedly incomplete) account of some exciting progress achieved since April 1997. I am grateful to the referee for helpful suggestions.

Theorem 1. LRr is a finitely generated subsemigroup of the additive semigroup Pr3 ⊂ ℤ3r. This is a special case of a much more general result well known to the experts in invariant theory. A short proof (valid for any reductive group instead of GLr(ℂ)) can be found in [Elashvili 1992]; A. Elashvili attributes this proof to M. Brion and F. Knop. The semigroup property also follows at once from “polyhedral” expressions for that will be discussed later (see Theorem 5 and below).