A Self-Dual Hopf Algebra on Double Partially Ordered Sets

Abstract

Let $\mathbf{D}$ be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On $\BZ\mathbf D$ we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations $\BZ\mathbf D$ is a Hopf algebra, self-dual with respect to a scalar product which counts the number of pictures (in the sense of Zelevinsky) between two double posets. 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 obtain a sub-Hopf-algebra 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.

Cited by in corpus (9)

• Decomposition spaces and restriction species
• Double posets and the antipode of QSym
• Free and cofree Hopf algebras
• Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials
• Algebraic structures on double and plane posets
• Surjections and double posets
• The Bernstein homomorphism via Aguiar-Bergeron-Sottile universality
• A theory of pictures for quasi-posets
• Plane posets, special posets, and permutations