[PAST EVENT] Mathematics Colloquium - Nicholas Mayers (NC State)
Access & Features
- Free food
- Open to the public
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.