# An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

What are the roles that the classic number arrays-- Eulerian, Narayana--play in the application of totally non-negative Grassmannians, or amplituhedrons, to string / twistor scattering theory?

(This is a severely reduced version of the original post, which can be found at my blog https://tcjpn.wordpress.com/2016/11/20/an-intriguing-tapestry-number-triangles-polytopes-grassmannians-and-scattering-amplitudes/, with numerous references to the combinatorics of the associahedra and permutohedra and relations to scattering theory and Grassmannians.)

• – Tom Copeland Nov 10 '16 at 21:27
• Resurrected at my website for easy access to and updating of refs: tcjpn.wordpress.com/2016/11/20/… – Tom Copeland Jun 29 '17 at 19:01
• See Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet by Arkani-Hamid et al. (arxiv.org/abs/1711.09102) The associahedra and its dual, discussed in the paper, are related to the Narayana numbers (oeis.org/A001263). – Tom Copeland Mar 10 '18 at 22:39
• My original post dealt with the combinatorics of the associahedra. – Tom Copeland Dec 10 '19 at 23:36
• This is directly related to the color-kinematics duality and the KLT relations connecting several quantum field theories – Tom Copeland Dec 11 '19 at 14:58

A bit long for a comment.

I like this question because as far as I know there are a lot of open problems concerned with expressing the links you mentioned via combinatorics. Perhaps you'd be interested in the following: "On some combinatorial and algebraic properties of Dunkl elements" - Anatol Kirillov.

The basic idea in that paper is that certain specializations and multiparameter deformations of Schubert and Grothendieck polynomials can be expressed through Narayana numbers, Catalan-Hankel determinants, and Schroder numbers, including some generalizations as well. In addition, these specializations come up in other areas such as $k$-triangulations of convex polygons and Carlitz-Riordan numbers. Through the wonderous abstract nonsense of Flags and degenerate loci, you can perhaps glean some knowledge.

• Thanks for the intriguing lead. A more current paper by Kirillov is "On some quadratic algebras, Dunkl elements, Schubert, Grothendieck, Tutte and reduced polynomials" at kurims.kyoto-u.ac.jp/preprint/file/RIMS1799.pdf – Tom Copeland Oct 9 '14 at 3:15
• That one is not for the faint of heart :). – Alex R. Oct 9 '14 at 4:21

In this recent preprint of Postnikov https://arxiv.org/abs/1806.05307 (which is for an upcoming ICM talk in Rio) he explains a beautiful direct connection between the geometry of the positive Grassmannian and the combinatorics of hypersimplices (polytopes whose normalized volume are the Eulerian numbers).

• Nice, building on work with his colleagues Speyer and Williams, noted in the original question. – Tom Copeland Jun 24 '18 at 20:12
• See specifically Theorem 11.1: "The poset of complete reduced Grassmannian graphs of type $(k, n)$ ordered by refinement is canonically isomorphic to the Baues poset $\omega(k, n, 2)$ of $\pi$-induced subdivisions for a $2$-dimensional cyclic projection $\pi$ of the hypersimplex $\Delta_{k,n}$. Under this isomorphism, plabic graphs correspond to tight $\pi$-induced subdivisions and moves of plabic graphs correspond to flips between tight $\pi$-induced subdivisions." – Sam Hopkins Jun 24 '18 at 20:13
• The Postnikov paper is "Positive Grassmannian and polyhedral subdivisions." – Tom Copeland Jun 25 '18 at 12:43
• These observations (as well as the note on permutohedra in the preprint) might well lead to geometric interpretations of the relations among the e.g.f.s and o.g.f.s presented in oeis.org/A046802 and oeis.org/A248727, related to enumerating positroids / Grassmannians, the f- and h-polynomials of the stellahedra / stellohedra, and the Eulerian numbers. – Tom Copeland Jun 25 '18 at 21:45

From 4gravitons' Post on the Amplitudes 2017 Conference: "Between the two of them, Nima (Arkani-Hamed) and Yuntao (Bai) covered an interesting development, tying the Amplituhedron together with the string theory-esque picture of scattering amplitudes pioneered by Freddy Cachazo, Song He, and Ellis Ye Yuan (or CHY). There’s a simpler (and older) Amplituhedron-like object called the associahedron that can be thought of as what the Amplituhedron looks like on the surface of a string, and CHY’s setup can be thought of as a sophisticated map that takes this object and turns it into the Amplituhedron."

The coarse h-polynomials of the associahedra (and their simplicial duals) are the Narayana polynomials A001263 while the refined h-partition polynomials are the noncrossing partitions A134264. The refined f-partition polynomials or face-partition polynomials of the associahedra are given by A133437 (unsigned and renormalized).

The permutohedra, whose h-polynomials are the Eulerian polynomials A008292 and f-partition polynomials A049019, are also discussed in (1) below.

Associated references are

(1) "Scattering forms and the positive geometry of kinematics, color, and the worldsheet" by N. Arkani-Hamed, Y. Bai, S. He, and G. Yan

(2) "Labelled tree graphs, Feynman diagrams and disk integrals" by X. Gao , S. He , Y. Zhang

(3) "Scattering from geometries" by S. He.

From (1):

"The search for a theory of the S-Matrix has revealed surprising geometric structures underlying amplitudes ranging from the worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to kinematic space where amplitudes live. In this paper, we propose a novel geometric understanding of amplitudes for a large class of theories. The key idea is to think of amplitudes not as functions, but rather as differential forms on kinematic space. We explore the resulting picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint $$\phi~^3$$ in scalar theory, we establish a direct connection between its “scattering form” and a classic polytope—the associahedron—known to mathematicians since the 1960’s. We find an associahedron living naturally in kinematic space, and the tree level amplitude is simply the “canonical form” associated with this “positive geometry”. Fundamental physical properties such as locality and unitarity, as well as novel “soft” limits, are fully determined by the combinatorial geometry of this polytope. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between the interior of this old “worldsheet associahedron” and the new “kinematic associahedron”, providing a geometric interpretation and simple conceptual derivation of the bi-adjoint CHY formula."