The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.

中文翻译：

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幅面体的奇偶对偶性

（树）振幅面体 $\mathcal {A}_{n,k,m}(Z)$ 是 Arkani-Hamed 和 Trnka 在 2013 年为了研究 $N=4 中的散射振幅而引入的 Grassmannian 的某个子集$ 超对称杨-米尔斯理论。证实第一作者的猜想，我们证明当 $m$ 是偶数时，仿射排列的集合对任何 $Z\ 产生 $\mathcal {A}_{n,k,m}(Z)$ 的三角剖分在 \operatorname {Gr}_{>0}(k+m,n)$ 当且仅当它们的逆集合产生 $\mathcal {A}_{n,nmk,m}(Z)$ 的三角剖分对于任何 $Z\in \operatorname {Gr}_{>0}(nk,n)$。我们使用 Marsh 和 Scott 的扭曲图来证明这种对偶性。我们还表明，该图保留了与相应正类细胞相关的规范微分形式，因此获得了幅面体微分形式的奇偶对偶性。