Well-Behaved Kohnert Posets

Dr. Nicholas Mayers
Postdoctoral Research Scholar
North Carolina State University

Kohnert polynomials form a family of polynomials indexed by diagrams that consist of unit cells arranged in the first quadrant. Many families of well-known polynomials have been shown to be examples of Kohnert polynomials, including key, Schur, and Schubert polynomials. Given a diagram D, the monomials occurring in the corresponding Kohnert polynomial encode diagrams formed from D by applying sequences of certain moves, called “Kohnert moves,” each of which alters the position of at most one cell. In this talk, we focus on the underlying sets of diagrams which generate the monomials of Kohnert polynomials. With each such collection of diagrams, one can associate a poset structure which is known to not, in general, be well-behaved. In particular, the corresponding “Kohnert posets” generally do not have a unique minimal element, are not ranked, and are not lattices. Here, we will focus on recent attempts to find conditions under which Kohnert posets are well-behaved in the sense that they have a unique minimal element, are ranked, or are EL-Shellable. No background knowledge concerning posets is assumed.