## Minuscule elements of Weyl groups

**John Stembridge**

(U. Michigan)

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.