Exterior Cyclic Polytope & Amplituhedron Connection

• Exterior cyclic polytope is a projective convex hull constructed via exterior products of ordered points, generalizing classical cyclic polytopes.
• It bridges the combinatorics of Grassmann geometry and amplituhedra with explicit f-vectors and detailed facet classifications in low-dimensional cases.
• Its study reveals duality properties, linking twist maps and scattering amplitude structures, which offer new insights into both theory and applications.

An exterior cyclic polytope is a projective polytope constructed as the convex hull of all exterior $k$-products of $n$ ordered points in projective space, generalizing the classical cyclic polytope. It simultaneously arises as the convex hull of the amplituhedron in its Plücker embedding, thereby establishing a deep connection between polytope combinatorics, Grassmann geometry, and scattering amplitudes. The theory encompasses duality under the projective dual, explicit combinatorics, connections to classical and dual cyclic polytopes, and relates to the matroidal and face-lattice structure of amplitude bodies and their convex hulls.

1. Formal Definition and Construction

Let $Z\in \mathrm{Mat}_{>0}(k+m,n)$ be a real $(k+m)\times n$ matrix with all maximal minors strictly positive; its columns $Z_1, \dots, Z_n$ are points in projective space $\mathbb{P}^{k+m-1}$. The classical cyclic polytope $C_{k+m,n}(Z)\subset \mathbb{P}^{k+m-1}$ is the convex hull $C_{k+m,n}(Z) = \mathrm{conv}\{Z_1,\dots,Z_n\}$. For any $k\leq k+m$, the \emph{exterior cyclic polytope} is defined as

$$C_{k,m,n}(Z) := \mathrm{conv}\left\{ Z_{i_1} \wedge \dotsb \wedge Z_{i_k} : 1\leq i_1<\cdots <i_k\leq n \right\} \subset \mathbb{P}^{\binom{k+m}{k}-1},$$

where the wedge product takes values in the corresponding projectivized exterior power. Alternatively, $n$0 is the image of the standard projective simplex $n$1 under the linear projection $n$2.

This construction generalizes the classical case: for $n$3, $n$4 recovers the combinatorics and face structure of a cyclic polytope of dimension $n$5.

2. Geometric Embedding and Relationship with Amplituhedra

The amplituhedron $n$6 is defined as the image of the nonnegative Grassmannian $n$7 under the linear map $n$8. The convex hull of $n$9 in the Plücker embedding equals the exterior cyclic polytope:

$Z\in \mathrm{Mat}_{>0}(k+m,n)$0

A key result for $Z\in \mathrm{Mat}_{>0}(k+m,n)$1 is that the amplituhedron coincides with the intersection of the Grassmannian and the exterior cyclic polytope: $Z\in \mathrm{Mat}_{>0}(k+m,n)$2 This result establishes a precise linear convexity for the $Z\in \mathrm{Mat}_{>0}(k+m,n)$3 amplituhedron inside the appropriate projective space (Mazzucchelli et al., 23 Jul 2025).

3. Combinatorial Structure: Vertices, Facets, and Face Lattice

The vertices of $Z\in \mathrm{Mat}_{>0}(k+m,n)$4 are indexed by $Z\in \mathrm{Mat}_{>0}(k+m,n)$5-element subsets $Z\in \mathrm{Mat}_{>0}(k+m,n)$6, corresponding to wedge products $Z\in \mathrm{Mat}_{>0}(k+m,n)$7. In the case $Z\in \mathrm{Mat}_{>0}(k+m,n)$8, these correspond to lines determined by pairs among the $Z\in \mathrm{Mat}_{>0}(k+m,n)$9 points. The full combinatorial structure is governed by the matroid $(k+m)\times n$0 of $(k+m)\times n$1.

Facets of $(k+m)\times n$2 are associated to hyperplanes in $(k+m)\times n$3. In $(k+m)\times n$4, there are two classes of facets:

• Schubert facets: Hyperplanes whose intersection with $(k+m)\times n$5 is a Schubert divisor (lines meeting a fixed line); there are $(k+m)\times n$6 such facets.
• Non-Schubert facets: Hyperplanes arising from circuits in $(k+m)\times n$7.

For $(k+m)\times n$8, the facet-defining inequalities are explicitly given by the normal vectors $(k+m)\times n$9, leading to the condition

$Z_1, \dots, Z_n$0

The $Z_1, \dots, Z_n$1-vector for small $Z_1, \dots, Z_n$2 in $Z_1, \dots, Z_n$3 cases is as follows:

$Z_1, \dots, Z_n$4	$Z_1, \dots, Z_n$5-vector	Total facets
5	(10, 35, 55, 40, 12, 1)	12 facets
6	(15, 75, 143, 111, 30, 1)	30 facets
7	(21, 147, 328, 282, 82, 1)	82 facets

The face lattice extends the classical Gale evenness: for $Z_1, \dots, Z_n$6, it recovers the usual cyclic polytope facet structure. For higher $Z_1, \dots, Z_n$7, Schubert facets are indexed by cyclic intervals; non-Schubert facets arise from higher wedge-circuit structure.

4. Duality: Dual Exterior Cyclic Polytope and Amplituhedron Duality

If $Z_1, \dots, Z_n$8 is a convex polytope with inward normal vectors $Z_1, \dots, Z_n$9, its projective dual $\mathbb{P}^{k+m-1}$0 is the convex hull of these normals.

For $\mathbb{P}^{k+m-1}$1, the Schubert-only exterior cyclic polytope satisfies a duality relation involving the twist map $\mathbb{P}^{k+m-1}$2. Specifically,

$\mathbb{P}^{k+m-1}$3

where $\mathbb{P}^{k+m-1}$4. The vertices of $\mathbb{P}^{k+m-1}$5 correspond to Schubert normal directions for $\mathbb{P}^{k+m-1}$6.

The amplituhedron duality is encapsulated by

$\mathbb{P}^{k+m-1}$7

meaning the dual amplituhedron (defined from the dual of the convex hull) is again an amplituhedron, with the matrix $\mathbb{P}^{k+m-1}$8 replaced by its twist. This parity duality links $\mathbb{P}^{k+m-1}$9 to $C_{k+m,n}(Z)\subset \mathbb{P}^{k+m-1}$0 (Mazzucchelli et al., 23 Jul 2025).

5. Relation to Classical Cyclic Polytopes and Generalizations

The exterior cyclic construction generalizes the classical cyclic polytope. The cyclic polytope $C_{k+m,n}(Z)\subset \mathbb{P}^{k+m-1}$1 is the convex hull of $C_{k+m,n}(Z)\subset \mathbb{P}^{k+m-1}$2 points on the moment curve in $C_{k+m,n}(Z)\subset \mathbb{P}^{k+m-1}$3. When $C_{k+m,n}(Z)\subset \mathbb{P}^{k+m-1}$4 and $C_{k+m,n}(Z)\subset \mathbb{P}^{k+m-1}$5 arbitrary, $C_{k+m,n}(Z)\subset \mathbb{P}^{k+m-1}$6 retains the combinatorics of $C_{k+m,n}(Z)\subset \mathbb{P}^{k+m-1}$7. For general $C_{k+m,n}(Z)\subset \mathbb{P}^{k+m-1}$8, the exterior cyclic polytope is the convex hull under the projection $C_{k+m,n}(Z)\subset \mathbb{P}^{k+m-1}$9, providing a genuine extension of classical cyclic polytopes to the context of Grassmannians and amplituhedra.

In the context of fixed-angles polytopes of convex $C_{k+m,n}(Z) = \mathrm{conv}\{Z_1,\dots,Z_n\}$0-gons, there is a duality between these "polygon" polytopes and cyclic polytopes $C_{k+m,n}(Z) = \mathrm{conv}\{Z_1,\dots,Z_n\}$1 governed by majority-dominant angle conditions (odd $C_{k+m,n}(Z) = \mathrm{conv}\{Z_1,\dots,Z_n\}$2) or additional "dipole tie-breaking" constraints (even $C_{k+m,n}(Z) = \mathrm{conv}\{Z_1,\dots,Z_n\}$3) (Ramshaw et al., 2020).

6. Low-Dimensional Cases and Explicit Examples

For $C_{k+m,n}(Z) = \mathrm{conv}\{Z_1,\dots,Z_n\}$4 (any $C_{k+m,n}(Z) = \mathrm{conv}\{Z_1,\dots,Z_n\}$5), $C_{k+m,n}(Z) = \mathrm{conv}\{Z_1,\dots,Z_n\}$6 is the convex hull of normals to hyperplanes through $C_{k+m,n}(Z) = \mathrm{conv}\{Z_1,\dots,Z_n\}$7-tuples of the $C_{k+m,n}(Z) = \mathrm{conv}\{Z_1,\dots,Z_n\}$8 points, combinatorially a $C_{k+m,n}(Z) = \mathrm{conv}\{Z_1,\dots,Z_n\}$9-simplex. In $k\leq k+m$0, $k\leq k+m$1, one has a polytope in $k\leq k+m$2 with 12 facets and $k\leq k+m$3-vector $k\leq k+m$4. For $k\leq k+m$5, the polytope has 30 facets (21 Schubert, 9 non-Schubert) with $k\leq k+m$6-vector $k\leq k+m$7. In $k\leq k+m$8, non-Schubert hyperplanes dependent on $k\leq k+m$9's minors occur, emphasizing the richer circuit structure.

7. Applications and Open Directions

Exterior cyclic polytopes, as convex hulls of amplituhedra, serve as a bridge between algebraic and combinatorial geometry, positive Grassmannians, and the study of scattering amplitudes. Open directions include stratification of matrix parameter space by combinatorial type, connections with triangulations and tilings paralleling known amplituhedron triangulations, and analysis of canonical forms or volume formulas derived from dual polytopes (Mazzucchelli et al., 23 Jul 2025). The projective duality and explicit face-lattice structure suggest further interaction with oriented matroid theory, tropical geometry, and the emergent convex geometry of higher amplitude bodies.