April 28, 2023
Contact: Nicholas Russoniello
Summary
{{https://math.sciences.ncsu.edu/people/nmayers/, Nick Mayers}} (NCSU)
Full Description
Title: Kohnert posets
Abstract: Kohnert polynomials form a family of polynomials indexed by diagrams consisting of
unit boxes arranged in the first quadrant. Many families of well-known polynomials have been
shown to be examples of Kohnert polynomials, including Key polynomials, Schur polynomials,
and Schubert polynomials. Given a diagram D, the monomials occurring in the corresponding
Kohnert polynomial KD are defined by diagrams formed from D by applying certain moves, called
“Kohnert moves,” which alter the position of at most one box. In this talk, we will be focusing on
combinatorial questions related to the underlying sets of diagrams which generate the monomials
of a Kohnert polynomial KD, denoted KD(D). It is known that one can associate a poset structure
to KD(D) which is, in general, not “well-behaved.” In particular, the corresponding “Kohnert
posets” generally do not have a unique minimal element, are not ranked, and are not lattices.
Here, in addition to some enumerative results, we will focus on recent attempts to find conditions
for when Kohnert posets are well-behaved in the sense that they have a unique minimal element or
are ranked. This is ongoing work with undergrads at both NC State as well as William & Mary.
