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Parity duality for the amplituhedron

Part of: Special varieties Arithmetic rings and other special rings Polytopes and polyhedra

Published online by Cambridge University Press:  09 December 2020

Pavel Galashin  and
Thomas Lam
Pavel Galashin
Affiliation:
Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA90025, USAgalashin@math.ucla.edu
Thomas Lam
Affiliation:
Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI48109-1043, USAtfylam@umich.edu
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Abstract

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.

Keywords

amplituhedrontwist maptotal positivityGrassmannianpositroid cellsscattering amplitudescyclic polytopes

MSC classification

Primary: 14M15: Grassmannians, Schubert varieties, flag manifolds
Secondary: 13F60: Cluster algebras 52B99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 11 , November 2020 , pp. 2207 - 2262
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Copyright
© The Author(s) 2020

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Footnotes

This work was partially supported by the National Science Foundation under Grants No. DMS-1954121 (P.G.), No. DMS-1464693 (T.L.), and No. DMS-1953852 (T.L.).

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