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43
The Diagonal of the Stasheff polytope
"... We construct an Ainfinity structure on the tensor product of two Ainfinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AAinfinity based on the simplicial Stasheff polytope. The operad AAinfinity admits a coassociative ..."
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Cited by 52 (2 self)
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We construct an Ainfinity structure on the tensor product of two Ainfinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AAinfinity based on the simplicial Stasheff polytope. The operad AAinfinity admits a coassociative diagonal and the operad Ainfinity is a retract by deformation of it. We compare these constructions with analogous constructions due to SaneblidzeUmble and MarklShnider based on the BoardmanVogt cubical decomposition of the Stasheff polytope.
Diagonals on the Permutahedra, Multiplihedra and associahedra
 J. HOMOLOGY, HOMOTOPY AND APPL
, 2004
"... We construct an explicit diagonal ∆P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks ’ projection P → K [19] and its factorization through J. We introduce the notion of a permutahedral set Z and lift ∆P to a diagonal on Z. We show that the ..."
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Cited by 47 (10 self)
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We construct an explicit diagonal ∆P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks ’ projection P → K [19] and its factorization through J. We introduce the notion of a permutahedral set Z and lift ∆P to a diagonal on Z. We show that the double cobar construction Ω²C∗(X) is a permutahedral set; consequently ∆P lifts to a diagonal on Ω²C∗(X). Finally, we apply the diagonal on K to define the tensor product of A∞(co)algebras in maximal generality.
Lattice congruences, fans and Hopf algebras
 J. Combin. Theory Ser. A
"... Abstract. We give a unified explanation of the geometric and algebraic properties of two wellknown maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak or ..."
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Cited by 32 (12 self)
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Abstract. We give a unified explanation of the geometric and algebraic properties of two wellknown maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the MalvenutoReutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of patternavoidance. Applying these results, we build the MalvenutoReutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of noncommutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations. 1.
Structure of the LodayRonco Hopf algebra of trees
 J. Algebra
"... Abstract. Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of noncommutative symmetric functions in the MalvenutoReutenauer Hopf algebra of permutations factors through their Hopf ..."
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Cited by 25 (3 self)
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Abstract. Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of noncommutative symmetric functions in the MalvenutoReutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra. We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopfalgebraic structure. We also obtain a transparent proof of its isomorphism with the noncommutative ConnesKreimer Hopf algebra of Foissy, and show that this algebra is related to noncommutative symmetric functions as the (commutative) ConnesKreimer Hopf algebra is related to symmetric functions.
Polydiagonal compactifications of configuration spaces
 J. Algebraic Geom
"... Abstract. A smooth compactification X〈n 〉 of the configuration space of n distinct labeled points in a smooth algebraic variety X is constructed by a natural sequence of blowups, with the full symmetry of the permutation group Sn manifest at each stage. The strata of the normal crossing divisor at i ..."
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Cited by 23 (0 self)
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Abstract. A smooth compactification X〈n 〉 of the configuration space of n distinct labeled points in a smooth algebraic variety X is constructed by a natural sequence of blowups, with the full symmetry of the permutation group Sn manifest at each stage. The strata of the normal crossing divisor at infinity are labeled by leveled trees and their structure is studied. This is the maximal wonderful compactification in the sense of De Concini–Procesi, and it has a stratacompatible surjection onto the Fulton–MacPherson compactification. The degenerate configurations added in the compactification are geometrically described by polyscreens similar to the screens of Fulton and MacPherson. In characteristic 0, isotropy subgroups of the action of Sn on X〈n〉 are abelian, thus X〈n 〉 may be a step toward an explicit resolution of singularities of the symmetric products X n /Sn.
Quotients of the multiplihedron as categorified associahedra
 Homotopy, Homology and Appl
, 2008
"... Abstract. We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associah ..."
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Cited by 12 (4 self)
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Abstract. We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associahedra, as was previously assumed in both topological and categorical literature. They are given the new collective name composihedra. We point out how these polytopes are used to parameterize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We also present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence of
Equivariant Fiber Polytopes
 DOCUMENTA MATH.
, 2002
"... The equivariant generalization of Billera and Sturmfels’ fiber polytope construction is described. This gives a new relation between the associahedron and cyclohedron, a different natural construction for the type B permutohedron, and leads to a family of orderpreserving maps between the face latti ..."
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Cited by 12 (2 self)
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The equivariant generalization of Billera and Sturmfels’ fiber polytope construction is described. This gives a new relation between the associahedron and cyclohedron, a different natural construction for the type B permutohedron, and leads to a family of orderpreserving maps between the face lattice of the type B permutohedron and that of the cyclohedron
From left modules to algebras over an operad: application to combinatorial Hopf algebras
, 2006
"... The purpose of this paper is two fold: we study the behaviour of the forgetful functor from Smodules to graded vector spaces in the context of algebras over an operad and derive from this theory the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness resu ..."
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Cited by 8 (2 self)
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The purpose of this paper is two fold: we study the behaviour of the forgetful functor from Smodules to graded vector spaces in the context of algebras over an operad and derive from this theory the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for these Hopf algebras. Let O denote the forgetful functor from Smodules to graded vector spaces. Left modules over an operad P are treated as Palgebras in the category of Smodules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad P: the functor O sends Palgebras to Palgebras. If P is a Hopf operad the functor O sends Hopf Palgebras to Hopf Palgebras. If the operad P is regular one gets two different structures of Hopf Palgebras in the category of graded vector spaces. We develop the notion of unital infinitesimal Pbialgebra and prove freeness and cofreeness results for Hopf algebras built from Hopf operads. Finally, we prove that many combinatorial Hopf algebras arise from our theory, as Hopf algebras on the faces of the permutohedra and associahedra.
GEOMETRIC COMBINATORIAL ALGEBRAS: CYCLOHEDRON AND SIMPLEX
"... Abstract. In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the MalvenutoReutenauer algebra of permutations and the LodayRonco ..."
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Cited by 8 (4 self)
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Abstract. In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the MalvenutoReutenauer algebra of permutations and the LodayRonco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time, that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra with basis all the faces of the simplices.