When alcoved polytopes add

TL;DR

This work addresses when Minkowski sums of alcoved polytopes remain alcoved by proving that pairwise compatibility suffices to guarantee global compatibility, and that the type fan is determined by its 2D cones. It provides a complete graphical criterion for when sums of alcoved simplices are alcoved, reducing checks to restricted partitions of size at most six and connecting to pattern avoidance in cyclic orders. The authors show prominent examples—the associahedron, cyclohedron, and a new infinite family hat{D}_n—are alcoved within this framework. They further relate their combinatorial results to matroidal blade arrangements and the Dressian, linking alcoved polytopes to tropical geometry and applications in physics. Overall, the paper advances the understanding of the Minkowski structure of alcoved polytopes, their normal fans, and their connections to combinatorial and geometric structures arising in mathematics and physics.

Abstract

Alcoved polytopes are characterized by the property that all facet normal directions are parallel to the roots $e_i-e_j$. Unlike other prominent families of polytopes, like generalized permutahedra, alcoved polytopes are not closed under Minkowski sums. We nonetheless show that the Minkowski sum of a collection of alcoved polytopes is alcoved if and only if each pairwise sum is alcoved. This implies that the type fan of alcoved polytopes is determined by its two-dimensional cones. Moreover, we provide a complete characterization of when the Minkowski sum of alcoved simplices is again alcoved via a graphical criterion on pairs of ordered set partitions. Our characterization reduces to checking conditions on restricted partitions of length at most six. In particular, we show how the Minkowski sum decompositions of the two most well-known families of alcoved polytopes, the associahedron and the cyclohedron, fit in our framework. Additionally, inspired by the physical construction of one-loop scattering amplitudes, we present a new infinite family of alcoved polytopes, called $\widehat{D}_n$ polytopes. We conclude by drawing a connection to matroidal blade arrangements and the Dressian.

Figures (3)

• Figure 1: The graphs $\Gamma_\sigma$, $\Gamma_\tau$, and $\Gamma_{\sigma,\tau}$ for the root cones described in Example \ref{['ex:root_cones']}. A primitive alternating $4$-cycle of $\Gamma_{\sigma,\tau}$ is highlighted in bold.
• Figure 2: The graph $G_\mathbf{S}$ of the ordered set partition $\mathbf{S}=(1,23,4)$.
• Figure 3: Left: the blade given by $((1,2,3,4))$. Middle: the blade given by $((3,2,1,4))$. Their common intersection is the one-dimensional subspace spanned by $e_1-e_2+e_3-e_4$; it is depicted as the black arrows. Right: the superposition, i.e. the normal fan to the Minkowski sum of the two alcoved simplices. Modulo a linear transformation, the Newton polytope shown is the root polytope, the convex hull of all roots $e_i-e_j$.

Theorems & Definitions (80)

• definition 2.1
• proposition 2.2
• proof : Proof of Theorem \ref{['thm:A']}
• definition 2.3
• lemma 2.4
• proof
• lemma 2.5
• lemma 2.6
• proof
• lemma 2.7