Shellability of Kohnert posets

TL;DR

This paper investigates the shellability of Kohnert posets linked to Kohnert polynomials, establishing conditions under which these posets are shellable and relating shellability to polynomial properties.

Contribution

It identifies when Kohnert posets are (EL-)shellable and connects shellability with polynomial multiplicity freeness under certain diagram restrictions.

Findings

Determines shellability conditions for key Kohnert posets.

Links shellability to multiplicity freeness of Kohnert polynomials.

Provides combinatorial criteria for poset shellability.

Abstract

In this paper, we are concerned with identifying among the family of posets associated with Kohnert polynomials, those whose order complex has a certain combinatorial property. In particular, for numerous families of Kohnert polynomials, including key polynomials, we determine when the associated Kohnert posets are (EL-)shellable. Interestingly, under certain diagram restrictions, (EL-)shellability of a Kohnert poset is equivalent to multiplcity freeness of the associated Kohnert polynomial.