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Lagrange’s theorem for Hopf monoids in species
, 2012
"... Abstract. Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras, ..."
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Abstract. Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras,
HOPF MONOIDS FROM CLASS FUNCTIONS ON UNITRIANGULAR MATRICES
"... Abstract. We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal’s category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and I ..."
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Abstract. We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal’s category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species, with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices.
ON THE HADAMARD PRODUCT OF HOPF MONOIDS
"... Dedicated to the memory of JeanLouis Loday Abstract. Combinatorial structures which compose and decompose give rise to Hopf monoids in Joyal’s category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The ..."
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Dedicated to the memory of JeanLouis Loday Abstract. Combinatorial structures which compose and decompose give rise to Hopf monoids in Joyal’s category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free. The first main result is obtained by showing the existence of a oneparameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco which applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species which is intertwined by the free monoid functor with the Hadamard product. As an application of the first result, we deduce that the dimension sequence of a connected Hopf monoid satisfies the following condition: except for the first, all coefficients of the reciprocal of its generating function are nonpositive.
A Hopfpower Markov chain on compositions
"... Abstract. In a recent paper, Diaconis, Ram and I constructed Markov chains using the coproductthenproduct map of a combinatorial Hopf algebra. We presented an algorithm for diagonalising a large class of these “Hopfpower chains”, including the GilbertShannonReeds model of riffleshuffling of a ..."
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Abstract. In a recent paper, Diaconis, Ram and I constructed Markov chains using the coproductthenproduct map of a combinatorial Hopf algebra. We presented an algorithm for diagonalising a large class of these “Hopfpower chains”, including the GilbertShannonReeds model of riffleshuffling of a deck of cards and a rockbreaking model. A very restrictive condition from that paper is removed in my thesis, and this extended abstract focuses on one application of the improved theory. Here, I use a new technique of lumping Hopfpower chains to show that the Hopfpower chain on the algebra of quasisymmetric functions is the induced chain on descent sets under riffleshuffling. Moreover, I relate its right and left eigenfunctions to GarsiaReutenauer idempotents and ribbon characters respectively, from which I recover an analogous result of Diaconis and Fulman (2012) concerning the number of descents under riffleshuffling. Résumé. Dans un récent article avec Diaconis et Ram, nous avons construit des chaînes de Markov en utilisant une composition du coproduit et produit d’une algébre de Hopf combinatoire. Nous avons présenté un algorithme pour diagonaliser une large classe de ces “chaînes de Hopf puissance”, en particulier nous avons diagonalisé le modèle de GilbertShannonReeds de mélange de cartes en “riffle shuffle ” (couper en deux, puis intercaler) et un modèle de cassage de pierres. Dans mon travail de thèse, nous supprimons une condition très restrictive de cet article, et ce papier se concentre sur
HOPF MONOIDS IN THE CATEGORY OF SPECIES
"... Abstract. A Hopf monoid (in Joyal’s category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a selfcontained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We stu ..."
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Abstract. A Hopf monoid (in Joyal’s category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a selfcontained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We study several examples of this nature. We emphasize the central role played in the theory by the Tits algebra of set compositions. Its product is tightly knit with the Hopf monoid axioms, and its elements constitute universal operations on connected Hopf monoids. We study analogues of the classical Eulerian and Dynkin idempotents and discuss the PoincaréBirkhoffWitt and CartierMilnorMoore theorems for Hopf monoids.
ON THE HADAMARD PRODUCT OF HOPF MONOIDS MARCELO AGUIAR
"... Abstract. Combinatorial structures which compose and decompose give rise to Hopf monoids in Joyal’s category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is conn ..."
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Abstract. Combinatorial structures which compose and decompose give rise to Hopf monoids in Joyal’s category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free. The first main result is obtained by showing the existence of a oneparameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco which applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species which is intertwined by the free monoid functor with the Hadamard product. As an application of the first result, we deduce that the Boolean transform of the dimension sequence of a connected Hopf monoid is nonnegative.
Hopf algebras and Markov chains: Two . . .
, 2012
"... The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural “rockbreaking” process, and Markov chains on simplicial complexes. ..."
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The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural “rockbreaking” process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rockbreaking, an explicit description of the quasistationary distribution and sharp rates to absorption follow.
Cofree compositions of coalgebras
"... Abstract. We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a onesided Hopf algebra. These c ..."
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Abstract. We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a onesided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. We conclude by discussing these structures for compositions with bases the vertices of multiplihedra, composihedra, and hypercubes. Résumé.
GENERALIZED HOPF MODULES FOR BIMONADS
"... Abstract. Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler’s Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodulemonad ..."
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Abstract. Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler’s Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodulemonad and a algebracomonoid over it. As an application we obtain a generalization of another classical theorem from the Hopf algebra literature, due to Schneider, which itself is an extension of Sweedler’s result (to the setting of Hopf Galois extensions).