On nu Faces of Partial Alternating Sign Matrix Polytopes

TL;DR

The paper defines and analyzes the $(\nu / \lambda)$-partial ASM polytope PASM$(\nu / \lambda,m,n)$ as a natural partial analogue of the ASMCRY polytope, with connections to the Chan–Robbins–Yuen polytope and the $\nu$-Tamari lattice. It establishes a concrete inequality description, proves that PASM$(\nu / \lambda,m,n)$ is a face of the larger partial ASM polytope $\mathrm{PASM}(m,n)$, and shows that PASM$(\nu / \lambda,m,n)$ is integrally equivalent to the order polytope $\mathcal{O}(P(\nu / \lambda))$, hence to a flow polytope for strongly planar posets. This leads to explicit formulas for its volume and Ehrhart polynomial via the poset $P(\nu / \lambda)$ and connects to the Naruse hook-length formula for skew tableaux. In a special case, the construction recovers an explicit integral equivalence with the ASMCRY polytope, generalizing the known correspondences between ASM polytopes, order/flow polytopes, and skew-Young tableau enumeration.

Abstract

We define and study the $(ν/ λ)$-partial alternating sign matrix polytope, motivated by connections to the Chan-Robbins-Yuen polytope and the $ν$-Tamari lattice. We determine the inequality description and show this polytope is a face of the partial alternating sign matrix polytope of [Heuer, Striker 2022]. We show that the $(ν/ λ)$-partial ASM polytope is an order polytope and a flow polytope.

Figures (8)

• Figure 1: The matrix $M^{(5,3,3,1)}_{5,7}$ with $(5,3,3,1)$ shown in white. The border strip (Definition \ref{['def:border_strip']}) associated to $(5,3,3,1)$ is shown in grey.
• Figure 2: All matrices in $\mathrm{PASM}_{4,5}^{(4,2,2) /(3,1)}$ with $\mu$ between $\lambda=(3,1)$ and $\nu=(4,2,2)$, where $\mu$ is in violet. Note: $\lambda$ is green, the border strip associated to $\nu$ is in gray.
• Figure 3: Examples of basic sum-labelings, corresponding to the matrices in Figure \ref{['fig:4x5examples']}, where $\mu$ is between $\lambda=(3,1)$ and $\nu=(4,2,2)$.
• Figure 4: The union of the sum-labelings in Figure \ref{['fig:4x5examples']}, with $\mu$ between $\lambda=(3,1)$ and $\nu=(4,2,2)$.
• Figure 5: Left: $E_{(3,1)}$ is highlighted in red. Right: $E_{(4,2,2)}$ is highlighted in blue.

Theorems & Definitions (38)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: ASMCRY
• Corollary 1.4: ASMCRY
• Definition 2.1: HeuerStriker
• Definition 2.2: HeuerStriker
• Definition 2.3
• Lemma 2.4
• proof
• Definition 2.5