Extending the Tamari Lattice to Some Compositions of Species

Forcey, Stefan

doi:10.1007/978-3-0348-0405-9_10

Cited by 3 publications

(3 citation statements)

References 19 publications

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“…Besides the usual ordering of tubings by inclusion, there is an alternate partial order introduced by Forcey [11] and Ronco [33]. We describe the restriction of their poset to MTub(G).…”

Section: Graph Associahedramentioning

confidence: 99%

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Lattices from graph associahedra and subalgebras of the Malvenuto-Reutenauer algebra

Barnard¹,

McConville²

2018

Preprint

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The Malvenuto-Reutenauer algebra is a well-studied combinatorial Hopf algebra with a basis indexed by permutations. This algebra contains a wide variety of interesting sub Hopf algebras, in particular the Hopf algebra of plane binary trees introduced by Loday and Ronco. We compare two general constructions of subalgebras of the Malvenuto-Reutenauer algebra, both of which include the Loday-Ronco algebra. The first is a construction by Reading defined in terms of lattice quotients of the weak order, and the second is a construction by Ronco in terms of graph associahedra. To make this comparison, we consider a natural partial ordering on the maximal tubings of a graph and characterize those graphs for which this poset is a lattice quotient of the weak order.

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“…Besides the usual ordering of tubings by inclusion, there is an alternate partial order introduced by Forcey [11] and Ronco [33]. We describe the restriction of their poset to MTub(G).…”

Section: Graph Associahedramentioning

confidence: 99%

“…In [11], Forcey posed the problem of determining whether L G is a lattice for any graph G. This turns out to be false in general; cf. Section 6.1.…”

Section: Introductionmentioning

confidence: 99%

Lattices from graph associahedra and subalgebras of the Malvenuto-Reutenauer algebra

Barnard¹,

McConville²

2018

Preprint

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“…There is also an isomorphic, but not identical, left weak order. See, for example, Forcey's chapter [18] in this volume.) Having defined Coxeter diagrams and the weak order, we are prepared to state the general definition of Cambrian lattices.…”

Section: Cambrian Latticesmentioning

confidence: 99%

From the Tamari Lattice to Cambrian Lattices and Beyond

Reading

2012

Associahedra, Tamari Lattices and Related Structures

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Abstract. In this chapter, we trace the path from the Tamari lattice, via lattice congruences of the weak order, to the definition of Cambrian lattices in the context of finite Coxeter groups, and onward to the construction of Cambrian fans. We then present sortable elements, the key combinatorial tool for studying Cambrian lattices and fans. The chapter concludes with a brief description of the applications of Cambrian lattices and sortable elements to Coxeter-Catalan combinatorics and to cluster algebras. A map from permutations to triangulationsThe road from the Tamari lattice to Cambrian lattices starts with a simple map from the set S n+1 of permutations of {1, . . . , n + 1} to the set of triangulations of a convex polygon with n + 3 vertices. This map connects the Tamari lattice to the weak order on permutations, and opens the door to understanding the Tamari lattice in a broader lattice-theoretic context.One of the many realizations of the Tamari lattice is as a partial order on triangulations of a convex polygon. Specifically, take Q to be a convex (n + 3)-gon in the plane and identify the vertices of Q with the numbers 0, 1, . . . , n + 1, n + 2. We require that the vertices 0 and n + 2 be on a horizontal line, with 0 to the left and with all other vertices below that line. Furthermore, we require that the vertices 1 through n + 1 be placed so that, for all i from 0 to n + 1, the vertex i is strictly further left than the vertex i + 1. A correct construction of Q, for n = 7, is shown in the top-left picture of Figure 2 for the case n = 7.A triangulation of Q is a tiling of Q by triangles whose vertices are contained in the vertex set of Q. The triangulation is specified by the collection of n diagonals of Q appearing as edges of the triangles. A diagonal flip on a triangulation of Q is the operation of removing a diagonal of the triangulation to create a quadrilateral from two triangles, and then inserting the other diagonal of the quadrilateral to create a new triangulation. The Tamari lattice is a partial order on triangulations of Q whose cover relations are given by diagonal flips. The two triangulations in the cover differ by exactly one diagonal of Q, and the higher triangulation in the cover relation is the one in which this diagonal has larger slope. The Tamari lattice, for n = 3, is shown in Figure 1.This definition of the Tamari lattice highlights its connection to the associahedron. Since the vertices of the associahedron can be labeled by triangulations of a fixed convex polygon such that edges are given by diagonal flips, the Hasse diagram of the Tamari lattice is isomorphic to the 1-skeleton of the associahedron.

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Lattices from graph associahedra and subalgebras of the Malvenuto–Reutenauer algebra

Barnard

McConville

2020

Algebra Univers.

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Extending the Tamari Lattice to Some Compositions of Species

Cited by 3 publications

References 19 publications

Lattices from graph associahedra and subalgebras of the Malvenuto-Reutenauer algebra

Lattices from graph associahedra and subalgebras of the Malvenuto-Reutenauer algebra

From the Tamari Lattice to Cambrian Lattices and Beyond

Lattices from graph associahedra and subalgebras of the Malvenuto–Reutenauer algebra

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