7 research outputs found
A Littlewood-Richardson rule for Grassmannian Schubert varieties
We propose a combinatorial model for the Schubert structure constants of the
complete flag manifold when one of the factors is Grassmannian.Comment: 6 pages, 2 figures - Final versio
A combinatorial proof that Schubert vs. Schur coefficients are nonnegative
We give a combinatorial proof that the product of a Schubert polynomial by a
Schur polynomial is a nonnegative sum of Schubert polynomials. Our proof uses
Assaf's theory of dual equivalence to show that a quasisymmetric function of
Bergeron and Sottile is Schur-positive. By a geometric comparison theorem of
Buch and Mihalcea, this implies the nonnegativity of Gromov-Witten invariants
of the Grassmannian.Comment: 26 pages, several colored figure
Fomin-Greene monoids and Pieri operations
We explore monoids generated by operators on certain infinite partial orders.
Our starting point is the work of Fomin and Greene on monoids satisfying the
relations and
if Given such a monoid, the non-commutative
functions in the variables are shown to commute. Symmetric functions in
these operators often encode interesting structure constants. Our aim is to
introduce similar results for more general monoids not satisfying the relations
of Fomin and Greene. This paper is an extension of a talk by the second author
at the workshop on algebraic monoids, group embeddings and algebraic
combinatorics at The Fields Institute in 2012.Comment: 33 pages, this is a paper expanding on a talk given at Fields
Institute in 201
Schubert polynomials and -Schur functions (Extended abstract)
The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type by a Schur function can be understood from the multiplication in the space of dual -Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the -Bruhat order given by Bergeron-Sottile, along with certain operators associated to this order. On the other side, we connect this poset with a graph on dual -Schur functions given by studying the affine grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual -Schur functions which are analogous to the ones given for the Schubert vs. Schur problem