Celebrating Loday’s associahedron

Pilaud, Vincent; Santos, Francisco; Ziegler, Günter M.

doi:10.1007/s00013-023-01895-6

Cited by 4 publications

(1 citation statement)

References 179 publications

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“…An excellent overview is provided by [8]. One particular realization of the associahedron has been studied by many authors [24, 28, 30, 32] (see also [27]), including Loday, whose name is most often associated to it. The associahedra constructed in this way turn out to be generalized associahedra corresponding to the linear orientation of the

A_n

diagram.…”

Section: Introductionmentioning

confidence: 99%

ABHY Associahedra and Newton polytopes of F$F$‐polynomials for cluster algebras of simply laced finite type

Bazier‐Matte,

Chapelier‐Laget,

Douville

et al. 2023

Journal of London Math Soc

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A new construction of the associahedron was recently given by Arkani‐Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the ‐polynomials of the corresponding cluster algebras. In addition, we show that the toric variety associated to the ‐vector fan has the property that its nef cone is simplicial.

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A_n

diagram.…”

Section: Introductionmentioning

confidence: 99%

ABHY Associahedra and Newton polytopes of F$F$‐polynomials for cluster algebras of simply laced finite type

Bazier‐Matte,

Chapelier‐Laget,

Douville

et al. 2023

Journal of London Math Soc

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Realizations of Multiassociahedra via Rigidity

Crespo Ruiz,

Santos

2024

Discrete Comput Geom

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Let $$\Delta _{k}(n)$$ Δ k ( n ) denote the simplicial complex of $$(k+1)$$ ( k + 1 ) -crossing-free subsets of edges in $${\left( {\begin{array}{c}[n]\\ 2\end{array}}\right) }$$ [ n ] 2 . Here $$k,n\in \mathbb {N}$$ k , n ∈ N and $$n\ge 2k+1$$ n ≥ 2 k + 1 . Jonsson (2003) proved that [neglecting the short edges that cannot be part of any $$(k+1)$$ ( k + 1 ) -crossing], $$\Delta _{k}(n)$$ Δ k ( n ) is a shellable sphere of dimension $$k(n-2k-1)-1$$ k ( n - 2 k - 1 ) - 1 , and conjectured it to be polytopal. The same result and question arose in the work of Knutson and Miller (Adv Math 184(1):161-176, 2004) on subword complexes. Despite considerable effort, the only values of (k, n) for which the conjecture is known to hold are $$n\le 2k+3$$ n ≤ 2 k + 3 (Pilaud and Santos, Eur J Comb. 33(4):632–662, 2012. https://doi.org/10.1016/j.ejc.2011.12.003) and (2, 8) (Bokowski and Pilaud, On symmetric realizations of the simplicial complex of 3-crossing-free sets of diagonals of the octagon. In: Proceedings of the 21st annual Canadian conference on computational geometry, 2009). Using ideas from rigidity theory and choosing points along the moment curve we realize $$\Delta _{k}(n)$$ Δ k ( n ) as a polytope for $$(k,n)\in \{(2,9), (2,10) , (3,10)\}$$ ( k , n ) ∈ { ( 2 , 9 ) , ( 2 , 10 ) , ( 3 , 10 ) } . We also realize it as a simplicial fan for all $$n\le 13$$ n ≤ 13 and arbitrary k, except the pairs (3, 12) and (3, 13). Finally, we also show that for $$k\ge 3$$ k ≥ 3 and $$n\ge 2k+6$$ n ≥ 2 k + 6 no choice of points can realize $$\Delta _{k}(n)$$ Δ k ( n ) via bar-and-joint rigidity with points along the moment curve or, more generally, via cofactor rigidity with arbitrary points in convex position.

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Rectangulotopes

Cardinal,

Pilaud

2025

European Journal of Combinatorics

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Celebrating Loday’s associahedron

Cited by 4 publications

References 179 publications

ABHY Associahedra and Newton polytopes of F$F$‐polynomials for cluster algebras of simply laced finite type

ABHY Associahedra and Newton polytopes of F$F$‐polynomials for cluster algebras of simply laced finite type

Realizations of Multiassociahedra via Rigidity

Rectangulotopes

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