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Hopf algebras and dendriform structures arising from parking functions

Jean-Christophe NovelliJean-Yves Thibon — 2007

Fundamenta Mathematicae

We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension ( n + 1 ) n - 1 in degree n). This algebra can be embedded into a noncommutative polynomial algebra in infinitely many variables. We determine its structure, and show that it admits natural quotients and subalgebras whose graded components have dimensions respectively given by the Schröder numbers (plane trees), the Catalan numbers, and powers of 3. These smaller algebras are always bialgebras and belong to some family...

Natural endomorphisms of quasi-shuffle Hopf algebras

Jean-Christophe NovelliFrédéric PatrasJean-Yves Thibon — 2013

Bulletin de la Société Mathématique de France

The Hopf algebra of word-quasi-symmetric functions ( 𝐖𝐐𝐒𝐲𝐦 ), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on 𝐖𝐐𝐒𝐲𝐦 . This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret 𝐖𝐐𝐒𝐲𝐦 as a convolution algebra of linear endomorphisms of quasi-shuffle...

Twisted action of the symmetric group on the cohomology of a flag manifold

Alain LascouxBernard LeclercJean-Yves Thibon — 1996

Banach Center Publications

Classes dual to Schubert cycles constitute a basis on the cohomology ring of the flag manifold F, self-adjoint up to indexation with respect to the intersection form. Here, we study the bilinear form (X,Y) :=〈X·Y, c(F)〉 where X,Y are cocycles, c(F) is the total Chern class of F and〈,〉 is the intersection form. This form is related to a twisted action of the symmetric group of the cohomology ring, and to the degenerate affine Hecke algebra. We give a distinguished basis for this form, which is a...

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