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

Malvenuto, Claudia et Reutenauer, Christophe (2011). « A self paired Hopf algebra on double posets and a Littlewood Richardson rule ». Journal of Combinatorial Theory, Series A, 118(4), pp. 1322-1333.

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Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra. We define a symmetric bilinear form on this Hopf algebra: it counts the number of pictures (in the sense of Zelevinsky) between two double posets. This form is a Hopf pairing, which means that product and coproduct are adjoint each to another. The product and coproduct correspond respectively to disjoint union of posets and to a natural decomposition of a poset into order ideals. Restricting to special double posets (meaning that the second order is total), we obtain a notion equivalent to Stanley's labelled posets, and a Hopf subalgebra already considered by Blessenohl and Schocker. The mapping which maps each double poset onto the sum of the linear extensions of its first order, identified via its second (total) order with permutations, is a Hopf algebra homomorphism, which is isometric and preserves the internal product, onto the Hopf algebra of permutations, previously considered by the two authors. Finally, the scalar product between any special double poset and double posets naturally associated to integer partitions is described by an extension of the Littlewood–Richardson rule.

Type: Article de revue scientifique
Mots-clés ou Sujets: Posets; Hopf algebras; Permutations; Quasi-symmetric functions; Littlewood–Richardson rule
Unité d'appartenance: Faculté des sciences > Département de mathématiques
Déposé par: Christophe Reutenauer
Date de dépôt: 28 avr. 2016 18:33
Dernière modification: 19 mai 2016 18:49
Adresse URL : http://archipel.uqam.ca/id/eprint/8363


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