A Markov growth process for Macdonald's distribution on reduced words

We give an algorithmic-bijective proof of Macdonald's reduced word identity in the theory of Schubert polynomials, in the special case where the permutation is dominant. Our bijection uses a novel application of David Little's generalized bumping algorithm. We also describe a Markov growth process for an associated probability distribution on reduced words. Our growth process can be implemented efficiently on a computer and allows for fast sampling of reduced words. We also discuss various partial generalizations and links to Little's work on the RSK algorithm.Comment: 16 pages, 5 figure

How to get the weak order out of a digraph ?

International audienceWe construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$$n-1$, $B$$n$, $Ã$$n$, and the flag weak order on the wreath product ℤ$r$ ≀ $S$$n$ introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$$n-1$ case, in which case we obtain the classical Stanley symmetric function.On construit une famille d’ensembles ordonnés à partir d’un graphe orienté, simple et acyclique munit d’une valuation sur ses sommets, puis on calcule les valeurs de leur fonction de Möbius respective. On montre que l’ordre faible sur les groupes de Coxeter $A$$n-1$, $B$$n$, $Ã$$n$, ainsi qu’une variante de l’ordre faible sur les produits en couronne ℤ$r$ ≀ $S$$n$ introduit par Adin, Brenti et Roichman (2012), sont des cas particuliers de cette construction. On conclura en expliquant brièvement comment ce travail peut-être utilisé pour définir des fonction quasi-symétriques, en insistant sur l’exemple de l’ordre faible sur $A$$n-1$ où l’on obtient les séries de Stanley classiques

A canonical expansion of the product of two Stanley symmetric functions

We study the problem of expanding the product of two Stanley symmetric functions F[subscript w]⋅F[subscript u] into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert F[subscript w] = lim[subscript n →∞] S[subscript 1[superscipt n]x w], and study the behavior of the expansion of S[subscript 1[superscript n] x w]⋅S[subscript 1[superscript n] x u] into Schubert polynomials as n increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability. We then study some other related stability properties, providing a second proof of the main result

Schubert complexes and degeneracy loci

Given a generic map between flagged vector bundles on a Cohen-Macaulay variety, we construct maximal Cohen-Macaulay modules with linear resolutions supported on the Schubert-type degeneracy loci. The linear resolution is provided by the Schubert complex, which is the main tool introduced and studied in this paper. These complexes extend the Schubert functors of Kra\'skiewicz and Pragacz, and were motivated by the fact that Schur complexes resolve maximal Cohen-Macaulay modules supported on determinantal varieties. The resulting formula in K-theory provides a "linear approximation" of the structure sheaf of the degeneracy locus, which can be used to recover a formula due to Fulton.Comment: 23 pages, uses tabmac.sty; v2: corrected typos and added reference