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Hopf algebras and dendriform structures arising from parking functions

100%

Novelli J.-C., Thibon J.-Y.

Fundamenta Mathematicae

2007

tom 193

nr 3

189-241

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 of di- or trialgebras occurring in the works of Loday and Ronco. Moreover, the fundamental notion of parkization allows one to endow the set of parking functions of fixed length with an associative multiplication (different from the one coming from the Shi arrangement), leading to a generalization of the internal product of symmetric functions. Several of the intermediate algebras are stable under this operation. Among them, one finds the Solomon descent algebra but also a new algebra based on a Catalan set, admitting the Solomon algebra as a left ideal.

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

81%

Lascoux A., Leclerc B., Thibon J.-Y.

Banach Center Publications

1996

tom 36

nr 1

111-124

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 deformation of the usual basis of Schubert polynomials, and apply it to the computation of the Schubert cycle expansions of Chern classes of flag manifolds.

Ograniczanie wyników

1 Banach Center Publications

1 Fundamenta Mathematicae

2 Thibon J.-Y.

1 Lascoux A.

1 Leclerc B.

1 Novelli J.-C.

1 2007

1 1996

Wyniki wyszukiwania

Hopf algebras and dendriform structures arising from parking functions

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