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A survey of the higher Stasheff-Tamari orders. (English) Zbl 1280.52010

Müller-Hoissen, Folkert (ed.) et al., Associahedra, Tamari lattices and related structures. Tamari memorial Festschrift. Basel: Birkhäuser (ISBN 978-3-0348-0404-2/hbk; 978-3-0348-0405-9/ebook). Progress in Mathematics 299, 351-390 (2012).
The cyclic polytope \(\mathbb{C}(n,d)\) is the combinatorial polytope given by the convex hull of \(n\) points of the form \((t_i,t_i^2, t_i^3,\dots,t_i^d)\in\mathbb R^d\) where \(0\leq t_1<t_2<\dots<t_n\) is a strictly increasing sequence of \(n\) positive real numbers. Triangulations of \(\mathbb{C}(n,d)\) carry two poset structures \(\mathrm{HST}_1(n,d)\) and \(\mathrm{HST}_2(n,d)\) called higher Stasheff-Tamari orders. Both orders coincide for \(d=2\) with the order underlying the Tamari lattice defined on triangulations of convex polygons.
The paper surveys the state of the art concerning the two orders \(\mathrm{HST}_1(n,d)\) and \(\mathrm{HST}_2(n,d)\). The order \(\mathrm{HST}_1(n,d)\) appeared historically first in work of Kapranov and Voevodsky in the context of pasting schemes and can be defined combinatorially in terms of upward flips relating the two triangulations of \(\mathbb{C}(d+2,d)\). Since any pair of triangulations of \(\mathbb{C}(n,d)\) can be related by a sequence of bistellar flips (upward flips and their inverses), the order \(\mathrm{HST}_1(n,d)\) has unique minimal and maximal elements (Theorem 6.5). The order \(\mathrm{HST}_2(n,d)\), which is at least as strong as \(\mathrm{HST}_1(n,d)\), is defined by comparing (functions given as) last coordinates of piecewise linear maps \(\mathbb{C}(n,d)\longrightarrow\mathbb{C}(n,d+1)\) obtained by extending the obvious map on vertices piecewise linearly to simplices of a triangulation. The orders \(\mathrm{HST}_1(n,d)\) and \(\mathrm{HST}_2(n,d)\) coincide for \(d=0,1,2,3\) and there are no known examples where they differ. This leads to the question whether they are the same (Open Problem 3.3). Subsequent chapters of the paper discuss different encodings (by submersion sets, snug rectangles and non-interlacing separated \(d/2\)-faces), the lattice property (both orders are lattices for \(d\leq 3\) but neither \(\mathrm{HST}_2(9,4)\) nor \(\mathrm{HST}_2(10,5)\) are lattices), homotopy types and Möbius functions, connections to flip graph connectivity, diameters, subdivisions, the Baues Problem (concerning homotopy types of structures related to loop spaces) and its subsequent generalization by Billera, Kapranov and Sturmfels, connections with higher Bruhat orders and enumerative problems (which are largely open, even the number of triangulations of \(\mathbb{C}(n,3)\) is unknown in general).
For the entire collection see [Zbl 1253.00013].

MSC:

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
06A99 Ordered sets

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Online Encyclopedia of Integer Sequences:

Triangulations of 4-dimensional cyclic polytopes.