Kohnert posets and polynomials of northeast diagrams

TL;DR

The paper develops a comprehensive framework for Kohnert polynomials and their posets indexed by northeast diagrams, proving that $\mathcal{P}(D)$ equals the elementary-move poset $\mathcal{P}^{ele}(D)$ for such diagrams. It provides complete, polynomial-time criteria for when $\mathcal{P}(D)$ is bounded or ranked and for when $\mathfrak{K}_D$ is monomial multiplicity-free, with all criteria depending only on the diagram $D$. The results are specialized to lock diagrams, yielding explicit conditions on the row-weight vector $\alpha$ for MMF, rankedness, and boundedness, and connecting to lock polynomials and Demazure crystal structures. These contributions advance the combinatorial understanding of Kohnert models and their geometric and representation-theoretic connections.

Abstract

Kohnert polynomials and their associated posets are combinatorial objects with deep geometric and representation theoretic connections, generalizing both Schubert polynomials and type A Demazure characters. In this paper, we explore the properties of Kohnert polynomials and their posets indexed by northeast diagrams. We give separate classifications of the bounded, ranked, and multiplicity-free Kohnert posets for northeast diagrams, each of which can be computed in polynomial time with respect to the number of cells in the diagram. As an initial application, we specialize these classifications to simple criteria in the case of lock diagrams.

Figures (15)

• Figure 1: The diagram $D=\{(2,2),(3,1),(4,1),(4,2)\}$
• Figure 2: $\mathcal{P}(D)$ where $D=\{(2,1),(2,2),(3,1),(3,2)\}$
• Figure 3: Two diagrams
• Figure 4: The lock diagram $\reflectbox{$D$}(\alpha)$ for $\alpha = (1, 3, 1, 0 , 2)$ .
• Figure 5: Three diagrams with labelings.

Theorems & Definitions (62)

• Definition 1.1: AS22
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Definition 2.4