797 research outputs found
Hook formulas for skew shapes III. Multivariate and product formulas
We give new product formulas for the number of standard Young tableaux of
certain skew shapes and for the principal evaluation of the certain Schubert
polynomials. These are proved by utilizing symmetries for evaluations of
factorial Schur functions, extensively studied in the first two papers in the
series "Hook formulas for skew shapes" [arxiv:1512.08348, arxiv:1610.04744]. We
also apply our technology to obtain determinantal and product formulas for the
partition function of certain weighted lozenge tilings, and give various
probabilistic and asymptotic applications.Comment: 40 pages, 17 figures. This is the third paper in the series "Hook
formulas for skew shapes"; v2 added reference to [KO1] (arxiv:1409.1317)
where the formula in Corollary 1.1 had previously appeared; v3 Corollary 5.10
added, resembles published versio
Skew Schubert polynomials
We define skew Schubert polynomials to be normal form (polynomial)
representatives of certain classes in the cohomology of a flag manifold. We
show that this definition extends a recent construction of Schubert polynomials
due to Bergeron and Sottile in terms of certain increasing labeled chains in
Bruhat order of the symmetric group. These skew Schubert polynomials expand in
the basis of Schubert polynomials with nonnegative integer coefficients that
are precisely the structure constants of the cohomology of the complex flag
variety with respect to its basis of Schubert classes. We rederive the
construction of Bergeron and Sottile in a purely combinatorial way, relating it
to the construction of Schubert polynomials in terms of rc-graphs.Comment: 10 pages, 7 figure
Skew Divided Difference Operators and Schubert Polynomials
We study an action of the skew divided difference operators on the Schubert
polynomials and give an explicit formula for structural constants for the
Schubert polynomials in terms of certain weighted paths in the Bruhat order on
the symmetric group. We also prove that, under certain assumptions, the skew
divided difference operators transform the Schubert polynomials into
polynomials with positive integer coefficients.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
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