Brick polytopes of spherical subword complexes and generalized associahedra

Pilaud, Vincent; Stump, Christian

doi:10.1016/j.aim.2015.02.012

Cited by 50 publications

(99 citation statements)

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“…C. Hohlweg, C. Lange, and H. Thomas then provided a family of c-generalized associahedra having c-Cambrian fans as normal fans by removing certain hyperplanes from the permutahedron [27]. V. Pilaud and C. Stump recovered c-generalized associahedra by giving explicit vertex and hyperplane descriptions purely in terms of the subword complex approach introduced in the present paper [45].…”

Section: Generalized Associahedramentioning

confidence: 99%

“…This new approach brought up a large family of spherical subword complexes that are realizable as the boundary of a polytope. In particular, it provides a new perspective on generalized associahedra [45]. Unfortunately, this polytope fails to realize the multi-associahedron.…”

Section: Multi-associahedra Of Type Amentioning

confidence: 99%

“…The subword complex approach provides new perspectives and methods for finding polytopal realizations. In a subsequent paper, C. Stump and V. Pilaud obtain a geometric construction of a class of subword complexes containing generalized associahedra purely in terms of subword complexes [45]. Clusters [20,[47][48][49] Generalized associahedron [12,26,45,61] Multi-clusters (present paper) Generalized multi-associahedron (existence conjectured)…”

Section: Generalized Multi-associahedra and Polytopality Of Sphericalmentioning

confidence: 99%

“…In a subsequent paper, C. Stump and V. Pilaud study the geometry of subword complexes and use the theory developed in the present paper to describe the connections to Coxeter-sortable elements, and how to recover Cambrian fans, Cambrian lattices, and the generalized associahedra purely in terms of subword complexes [45].…”

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Subword complexes, cluster complexes, and generalized multi-associahedra

2013

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In this paper, we use subword complexes to provide a uniform approach to finite-type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k = 1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations, respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.

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Section: Generalized Associahedramentioning

confidence: 99%

Section: Multi-associahedra Of Type Amentioning

confidence: 99%

Section: Generalized Multi-associahedra and Polytopality Of Sphericalmentioning

confidence: 99%

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confidence: 99%

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Subword complexes, cluster complexes, and generalized multi-associahedra

2013

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“…Since then, the associahedron has motivated a flourishing research trend with rich connections to combinatorics, geometry and algebra: polytopal constructions [Lod04, HL07, CSZ15, LP13], Tamari and Cambrian lattices [MHPS12,Rea04,Rea06], diameter and Hamiltonicity [STT88,Deh10,Pou14,HN99], geometric properties [BHLT09,HLR10,PS15b], combinatorial Hopf algebras [LR98,HNT05,Cha00,CP14], to cite a few. The associahedron was also generalized in several directions, in particular to secondary and fiber polytopes [GKZ08,BFS90], graph associahedra and nestohedra [CD06,Dev09,Pos09,FS05,Zel06,Pil13], pseudotriangulation polytopes [RSS03], cluster complexes and generalized associahedra [FZ03b,CFZ02,HLT11,Ste13,Hoh12], and brick polytopes [PS12,PS15a].…”

Section: Introductionmentioning

confidence: 99%

Compatibility fans for graphical nested complexes

Manneville¹,

Pilaud²

2017

Journal of Combinatorial Theory, Series A

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Abstract. Graph associahedra are natural generalizations of the classical associahedra. They provide polytopal realizations of the nested complex of a graph G, defined as the simplicial complex whose vertices are the tubes (i.e. connected induced subgraphs) of G and whose faces are the tubings (i.e. collections of pairwise nested or non-adjacent tubes) of G. The constructions of M. Carr and S. Devadoss, of A. Postnikov, and of A. Zelevinsky for graph associahedra are all based on the nested fan which coarsens the normal fan of the permutahedron. In view of the combinatorial and geometric variety of simplicial fan realizations of the classical associahedra, it is tempting to search for alternative fans realizing graphical nested complexes.Motivated by the analogy between finite type cluster complexes and graphical nested complexes, we transpose in this paper S. Fomin and A. Zelevinsky's construction of compatibility fans from the former to the latter setting. For this, we define a compatibility degree between two tubes of a graph G. Our main result asserts that the compatibility vectors of all tubes of G with respect to an arbitrary maximal tubing on G support a complete simplicial fan realizing the nested complex of G. In particular, when the graph G is reduced to a path, our compatibility degree lies in {−1, 0, 1} and we recover F. Santos' Catalan many simplicial fan realizations of the associahedron.keywords. Graph associahedra · finite type cluster algebras · compatibility degrees · compatibility fans.MSC classes. 52B11, 52B12, 05E45.

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EL-Labelings and Canonical Spanning Trees for Subword Complexes

Pilaud

Stump

2013

Discrete Geometry and Optimization

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Abstract. We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facet-ridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these trees yields an efficient algorithm to generate all facets of the subword complex, which extends the greedy flip algorithm for pointed pseudotriangulations. On the other hand, when the increasing flip graph is a Hasse diagram, we show that the edge labeling is indeed an EL-labeling and derive further combinatorial properties of paths in the increasing flip graph. These results apply in particular to Cambrian lattices, in which case a similar EL-labeling was recently studied by M. Kallipoliti and H. Mühle.

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Brick polytopes of spherical subword complexes and generalized associahedra

Cited by 50 publications

References 42 publications

Subword complexes, cluster complexes, and generalized multi-associahedra

Subword complexes, cluster complexes, and generalized multi-associahedra

Compatibility fans for graphical nested complexes

EL-Labelings and Canonical Spanning Trees for Subword Complexes

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