Abstract: This talk will describe a combinatorial, probabilistic, and geometric perspective on the same object: the generating function for reverse plane partitions over a poset, i.e. for order reversing maps from a poset to the nonnegative integers. Such generating functions were considered at length in the PhD thesis of Richard Stanley, where several general results–including rationality and an explicit formula for the denominator–are proven. For special posets arising in representation theory called minuscule posets, these generating functions take an especially nice form and also arise in two other settings. First, they are the partition functions for the so-called Schur probabilistic processes defined by Okounkov-Reshitikhin. Second, they are the generating functions for degrees of quasimaps to certain Nakajima quiver varieties. I will explain some work in progress in which one of these three perspectives is used to shed light on another. For example, we can use quiver varieties to define new probabilistic processes, which conjecturally should have nice properties. As another direction, we can define a class of posets beyond minuscule posets for which Stanley’s generating function is just as nice.
