Dale Peterson has defined and studied what he calls "lambda-minuscule" elements of (symmetrizable Kac-Moody) Weyl groups. These elements can be encoded by, or even identified with, a certain class of labeled partially ordered sets. In type A, the posets are Young diagrams. In total, there are 17 "irreducible" families of these posets, 16 of which have infinitely many members.
As has become increasing clear in ongoing work of Robert Proctor, there is an amazingly rich combinatorial theory hidden in these posets, generalizing much of the classical combinatorics of Young diagrams. For example, there is an explicit product formula, due to Peterson (refined later by Proctor) for the number of reduced expressions for any lambda-minuscule element. This generalizes the famous hooklength formula of Frame-Robinson-Thrall for counting standard Young tableaux.
In this talk, we will survey the subject matter, including various characterizations, classifications, and applications.