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# Annales de l’Institut Henri Poincaré D

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**Volume 7, Issue 3, 2020, pp. 303–363**

**DOI: 10.4171/AIHPD/87**

Published online: 2020-08-22

Decompositions of amplituhedra

Steven N. Karp^{[1]}, Lauren K. Williams

^{[2]}and Yan X Zhang

^{[3]}(1) University of Michigan, Ann Arbor, USA

(2) University of California, Berkeley, USA

(3) San José State University, USA

The *(tree) amplituhedron* $\mathcal{A}_{n,k,m}$ is the image in the Grassmannian $\mathrm {Gr}_{k,k+m}$ of the totally nonnegative Grassmannian $\mathrm {Gr}_{k,n}^{\geq 0}$, under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in planar $\mathcal{N}=4$ supersymmetric Yang–Mills theory. In the case relevant to physics ($m=4$), there is a collection of recursively-defined $4k$-dimensional *BCFW cells* in $\mathrm {Gr}_{k,n}^{\geq 0}$, whose images conjecturally "triangulate" the amplituhedron – that is, their images are disjoint and cover a dense subset of $\mathcal{A}_{n,k,4}$. In this paper, we approach this problem by first giving an explicit (as opposed to recursive) description of the BCFW cells. We then develop sign-variational tools which we use to prove that when $k=2$, the images of these cells are disjoint in $\mathcal{A}_{n,k,4}$. We also conjecture that for arbitrary even $m$, there is a decomposition of the amplituhedron $\mathcal{A}_{n,k,m}$ involving precisely $M\big(k, n-k-m, \frac{m}{2}\big)$ top-dimensional cells (of dimension $km$), where $M(a,b,c)$ is the number of plane partitions contained in an $a \times b \times c$ box. This agrees with the fact that when $m=4$, the number of BCFW cells is the Narayana number $N_{n-3,k+1} = \frac{1}{n-3}\binom{n-3}{k+1}\binom{n-3}{k}$.

*Keywords: *Amplituhedron, scattering amplitude, totally nonnegative Grassmannian, BCFW recursion, Narayana number, plane partition

Karp Steven, Williams Lauren, Zhang Yan: Decompositions of amplituhedra. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 7 (2020), 303-363. doi: 10.4171/AIHPD/87