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On the face stratification of the $m=2$ amplituhedron

Thomas Lam

TL;DR

This work analyzes the face structure of the m=2 amplituhedron by embedding the nonnegative Grassmannian into the twistor space and intersecting with the positroid stratification of ${\rm Gr}(2,n)$. The authors construct a twistor map on matroids, identify the face poset $P_{n,k}$ as an upper order ideal in $Q_{n,2}$, and prove that the face stratification coincides with the semialgebraic/positive-geometry stratification, confirming predictions from Lukowski and related work. They give a concrete combinatorial description via pairs $(L,\{[a_i,b_i]\})$, derive a closed-form corank generating function, and establish the Eulerian nature of the augmented poset, together with dimension and cohomology computations that connect faces to Schubert calculus. Overall, the paper provides a complete combinatorial and geometric framework for the m=2 amplituhedron's faces, validating prior conjectures and offering tools for extending to broader m and k.

Abstract

We define and study the face stratification of the m=2 amplituhedron. We show that the face poset is an upper order ideal in the face poset of the totally nonnegative Grassmannian. Our construction is consistent with earlier work of Lukowski, and we confirm various predictions of Lukowski.

On the face stratification of the $m=2$ amplituhedron

TL;DR

This work analyzes the face structure of the m=2 amplituhedron by embedding the nonnegative Grassmannian into the twistor space and intersecting with the positroid stratification of . The authors construct a twistor map on matroids, identify the face poset as an upper order ideal in , and prove that the face stratification coincides with the semialgebraic/positive-geometry stratification, confirming predictions from Lukowski and related work. They give a concrete combinatorial description via pairs , derive a closed-form corank generating function, and establish the Eulerian nature of the augmented poset, together with dimension and cohomology computations that connect faces to Schubert calculus. Overall, the paper provides a complete combinatorial and geometric framework for the m=2 amplituhedron's faces, validating prior conjectures and offering tools for extending to broader m and k.

Abstract

We define and study the face stratification of the m=2 amplituhedron. We show that the face poset is an upper order ideal in the face poset of the totally nonnegative Grassmannian. Our construction is consistent with earlier work of Lukowski, and we confirm various predictions of Lukowski.
Paper Structure (9 sections, 23 theorems, 23 equations)

This paper contains 9 sections, 23 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. The amplituhedron in twistor space
  3. The twistor map on matroids
  4. The face stratification of the $m=2$ amplituhedron
  5. The poset $P_{n,k}$
  6. Lukowski matrices
  7. Rank-generating function
  8. Eulerian-ness
  9. Dimension estimates

Key Result

Lemma 1

For $Z$ full rank, the map $\psi: {\rm Gr}(k,k+m) \to {\rm Gr}(m,n)$ is an embedding.

Theorems & Definitions (57)

  • Lemma 1
  • proof
  • Proposition 2: GLconncomp
  • Remark 3
  • Theorem 4
  • Remark 5
  • Definition 6
  • Proposition 7
  • proof
  • Proposition 8
  • ...and 47 more