Supports of Castelnuovo-Mumford polynomials

TL;DR

The paper investigates when the supports of Castelnuovo-Mumford polynomials, the top-degree components of Grothendieck polynomials, form M-convex sets. By developing and refining bubbling-diagram and related combinatorial frameworks, it proves that for vexillary permutations the top component $\hat{\mathfrak{G}}_w$ is an integer multiple of the dual character $\chi_{D_{\text{top}}(w)}$, implying M-convex support and schubitope Newton polytopes. It extends these results to almost vexillary permutations and dominant fireworks-vexillary chains, showing their homogenized polynomials have M-convex supports as well, via new bubbling-diagram constructions and diagrammatic correspondences (e.g., $D_{\text{top}}$, snow diagrams). The work thus broadens the class of permutations for which Castelnuovo-Mumford supports are known to be M-convex, and provides a unified combinatorial toolkit to generate and verify such properties in future cases. Overall, the results offer concrete, diagrammatic mechanisms to certify M-convexity and connect representation-theoretic data to generalized permutahedra in Grothendieck polynomial theory.

Abstract

The Castelnuovo-Mumford polynomials are the maximal degree components of Grothendieck polynomials. The support of each Castelnuovo-Mumford polynomial is conjectured to be M-convex, i.e. the set of integer points of a generalized permutahedron (Mészáros and St. Dizier, 2020). This conjecture is known to hold in certain special cases but remains open in general. We define new families of permutations whose Castelnuovo-Mumford polynomials we show to have M-convex support. Specifically, we investigate which permutations have Castelnuovo-Mumford polynomials whose supports are the set of integer points in a schubitope.

Figures (9)

• Figure 1: Rothe diagram for $w= 18273564$. Numbers in each cell indicate the value of the rank function $r_{D(w)}(i,j)$.
• Figure 2: Skyline diagram for $\alpha=(0,6,0,4,0,1,1,0)$
• Figure 3: Left: Skyline diagram for $\alpha=(0,6,0,4,0,1,1,0)$. Right: Snow diagram $\text{snow}(D_{Sky}(\alpha))$ with dark cloud diagram shown in gray.
• Figure 4: A dead square diagram (left) along with the result of performing a bubbling move (center) and a K-bubbling move (right). Distinguished live squares are shown in gold and dead squares in gray.
• Figure 5: Top left: $D(w)$ for $w=1624735$ showing the left bubbling order and $A_L(w)$. Top right: $D(w)$ showing the right bubbling order and $A_R(w)$. Bottom left: $D_{\text{top}}(w)$. Bottom right: $D_{\text{top}}(D(w),A_R(w))$.

Theorems & Definitions (67)

• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Proposition 2.3: fink2018schubert*Theorem 7
• Definition 2.4
• Theorem 2.5: pan2024top, Theorem 1.2, Proof of Lemma 4.5
• Theorem 2.6: yu2023connection, Corollary 7.3
• Theorem 2.7: meszaros2022orthodontia,Theorem 1.1