The Magic Number Conjecture for the m=2 amplituhedron and Parke-Taylor identities
arxiv(2024)
摘要
The amplituhedron 𝒜_n,k,m is a geometric object introduced in
the context of scattering amplitudes in 𝒩=4 super Yang Mills. It
generalizes the positive Grassmannian (when n=k+m), cyclic polytopes (when
k=1), and the bounded complex of the cyclic hyperplane arrangement (when
m=1). Of substantial interest are the tilings of the amplituhedron, which are
analogous to triangulations of a polytope. Karp, Williams and Zhang (2020)
observed that the known tilings of 𝒜_n,k,2 have cardinality n-2
k and the known tilings of 𝒜_n,k,4 have cardinality the
Narayana number 1/n-3n-3 k+1n-3 k; generalizing
these observations, they conjectured that for even m the tilings of
𝒜_n, k,m have cardinality the MacMahon number, the number of
plane partitions which fit inside a k × (n-k-m) ×m/2 box.
We refer to this prediction as the `Magic Number Conjecture'. In this paper we
prove the Magic Number Conjecture for the m=2 amplituhedron: that is, we show
that each tiling of 𝒜_n,k,2 has cardinality n-2 k. We
prove this by showing that all positroid tilings of the hypersimplex
Δ_k+1,n have cardinality n-2 k, then applying T-duality.
In addition, we give combinatorial necessary conditions for tiles to form a
tiling of 𝒜_n,k,2; we give volume formulas for Parke-Taylor
polytopes and certain positroid polytopes in terms of circular extensions of
cyclic partial orders; and we prove new variants of the classical Parke-Taylor
identities.
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