A self paired Hopf algebra on double posets and a Littlewood–Richardson rule

Malvenuto, Claudia; Reutenauer, Christophe

doi:10.1016/j.jcta.2010.10.010

Cited by 27 publications

(44 citation statements)

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“…Inspired by Stanley's labelled posets, Malvenuto and Reutenauer [28] introduced double poset in the context of combinatorial Hopf algebras. A double poset P is a triple consisting of a finite ground set P and two partial order relations + and − on P .…”

Section: Introductionmentioning

confidence: 99%

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Two Double Poset Polytopes

Chappell¹,

Friedl²,

Sanyal³

2017

SIAM J. Discrete Math.

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Abstract. To every poset P , Stanley (1986) associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of P . This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer (2011) introduced double posets, that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labelled posets. Many combinatorial constructions can be naturally phrased in terms of double posets. We introduce the double order polytope and the double chain polytope and we amply demonstrate that they geometrically capture double posets, i.e., the interaction between the two partial orders. We describe the facial structures, Ehrhart polynomials, and volumes of these polytopes in terms of the combinatorics of double posets. We also describe a curious connection to Geissinger's valuation polytopes and we characterize 2-level polytopes among our double poset polytopes.Fulkerson's anti-blocking polytopes from combinatorial optimization subsume stable set polytopes of graphs and chain polytopes of posets. We determine the geometry of Minkowskiand Cayley sums of anti-blocking polytopes. In particular, we describe a canonical subdivision of Minkowski sums of anti-blocking polytopes that facilitates the computation of Ehrhart (quasi-)polynomials and volumes. This also yields canonical triangulations of double poset polytopes.Finally, we investigate the affine semigroup rings associated to double poset polytopes. We show that they have quadratic Gröbner bases, which gives an algebraic description of the unimodular flag triangulations described in the first part.

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Section: Introductionmentioning

confidence: 99%

“…This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer (2011) introduced double posets, that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labelled posets. Many combinatorial constructions can be naturally phrased in terms of double posets.…”

mentioning

confidence: 99%

Two Double Poset Polytopes

Chappell¹,

Friedl²,

Sanyal³

2017

SIAM J. Discrete Math.

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“…It follows that D → C is a right fibration, and hence a directed restriction species. The associated incidence coalgebra was first studied by Malvenuto and Reutenauer [39]; see [13] and [14] for more recent developments.…”

Section: Examples: Double Posets and Related Structuresmentioning

confidence: 99%

Decomposition Spaces and Restriction Species

Gálvez-Carrillo

Kock

Tonks

2018

International Mathematics Research Notices

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We show that Schmitt's restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital 2-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce the notion of directed restriction species that subsume Schmitt's restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and convex maps. We also introduce the notion of monoidal (directed) restriction species, which induce monoidal decomposition spaces and hence bialgebras, most often Hopf algebras. Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. A prominent instance of a resulting incidence bialgebra is the Butcher-Connes-Kreimer Hopf algebra of rooted trees. Both ordinary and directed restriction species are shown to be examples of a construction of decomposition spaces from certain cocartesian fibrations over the category of finite ordinals that are also cartesian over convex maps. The proofs rely on some beautiful simplicial combinatorics, where the notion of convexity plays a key role. The methods developed are of independent interest as techniques for constructing decomposition spaces. Proposition 7.9. Monoidal directed restriction species naturally induce monoidal decomposition spaces and hence bialgebras.

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“…The existence of internal products (by which we mean the existence of an associative product on double posets with a given cardinality) is a classical property of combinatorial Hopf algebras: in the representation theory of the symmetric group (or equivalently in the algebra of symmetric functions) the internal product is obtained from the tensor product of representations, and this product extends naturally to various noncommutative versions, such as the descent algebra or the Malvenuto-Reutenauer Hopf algebra [15].…”

Section: Internal Productsmentioning

confidence: 99%

“…Let us point out in particular recent developments (motivated by applications to multiple zeta values, Rota-Baxter algebras, stochastic integrals... [4,2,3]) that extend to surjections [18,17,14,13] the theory of combinatorial Hopf algebra structures on permutations [15,7]. New results on surjections will be obtained in the last section of the article.…”

Section: Introductionmentioning

confidence: 99%

A theory of pictures for quasi-posets

2017

Self Cite

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Abstract. The theory of pictures between posets is known to encode much of the combinatorics of symmetric group representations and related topics such as Young diagrams and tableaux. Many reasons, combinatorial (e.g. since semi-standard tableaux can be viewed as double quasi-posets) and topological (quasi-posets identify with finite topologies) lead to extend the theory to quasi-posets. This is the object of the present article. IntroductionThe theory of pictures between posets is known to encode much of the combinatorics of symmetric group representations and related topics such as preorder diagrams and tableaux. The theory captures for example the Robinson-Schensted (RS) correspondence or the Littlewood-Richardson formula, as already shown by Zelevinsky in the seminal article [20]. Recently, the theory was extended to double posets (pairs of orders coexisting on a given finite set -hereafter, "order" means "partial order"; an order on X defines a poset structure on X) and developed from the point of view of combinatorial Hopf algebras which led to new advances in the field [16, 8, 9, 10].In applications, a fundamental property that has not been featured enough, is that often pictures carry themselves implicitly a double poset structure. A typical example is given by standard Young tableaux, which can be put in bijection with certain pictures (this is one of the nicest way in which their appearance in the RS correspondence can be explained [20]) and carry simultaneously a poset structure (induced by their embeddings into N × N equipped with the coordinate-wise partial order) and a total order (the one induced by the integer labelling of the entries of the tableaux).However, objects such as tableaux with repeated entries, such as semistandard tableaux, although essential, do not fit into this framework. They should actually be thought of instead as double quasi-posets (pairs of preorders on a given finite set): the first preorder is the same than for standard tableaux (it is an order), but the labelling by (possibly repeated) integers is naturally captured by a preorder on the entries of the tableau (the one for which two entries are equivalent if they have the same label and else are ordered according to their labels).Besides the fact that these ideas lead naturally to new results and structures on preorders, other observations and motivations have led us to develop on systematic bases in the present article a theory of pictures for quasi-posets. Let us point out in particular recent developments (motivated by applications to multiple zeta values, Rota-Baxter algebras, stochastic integrals... [4,2,3]) that extend to surjections [18,17,14,13] the theory of combinatorial Hopf algebra structures on permutations [15,7]. New results on surjections will be obtained in the last section of the article.Lastly, let us mention our previous works on finite topologies (equivalent to quasi-posets) [11,12] (see also [5, 6] for recent developments) which featured the two products defined on finite topologies by disjoint unio...

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A self paired Hopf algebra on double posets and a Littlewood–Richardson rule

Cited by 27 publications

References 20 publications

Two Double Poset Polytopes

Two Double Poset Polytopes

Decomposition Spaces and Restriction Species

A theory of pictures for quasi-posets

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