750 research outputs found
Specializations of Grothendieck polynomials
We prove a formula for double Schubert and Grothendieck polynomials
specialized to two rearrangements of the same set of variables. Our formula
generalizes the usual formulas for Schubert and Grothendieck polynomials in
terms of RC-graphs, and it gives immediate proofs of many other important
properties of these polynomials.Comment: 4 pages, 1 figur
Maximal 0-1 fillings of moon polyominoes with restricted chain-lengths and rc-graphs
We show that maximal 0-1-fillings of moon polynomials with restricted chain
lengths can be identified with certain rc-graphs, also known as pipe dreams. In
particular, this exhibits a connection between maximal 0-1-fillings of Ferrers
shapes and Schubert polynomials. Moreover, it entails a bijective proof showing
that the number of maximal fillings of a stack polyomino S with no north-east
chains longer than k depends only on k and the multiset of column heights of S.
Our main contribution is a slightly stronger theorem, which in turn leads us to
conjecture that the poset of rc-graphs with covering relation given by
generalised chute moves is in fact a lattice.Comment: 22 pages, v2: references added, v3: included proof for bijection for
stack polyominoes, v4: include conjecture and improve presentatio
Pipe dreams for Schubert polynomials of the classical groups
Schubert polynomials for the classical groups were defined by S.Billey and
M.Haiman in 1995; they are polynomial representatives of Schubert classes in a
full flag variety of a classical group. We provide a combinatorial description
for these polynomials, as well as their double versions, by introducing
analogues of pipe dreams, or RC-graphs, for the Weyl groups of the classical
types.Comment: 36 pages, 10 figures. v2: appendix adde
Mitosis recursion for coefficients of Schubert polynomials
Mitosis is a rule introduced by [Knutson-Miller, 2002] for manipulating
subsets of the n by n grid. It provides an algorithm that lists the reduced
pipe dreams (also known as rc-graphs) [Fomin-Kirillov, Bergeron-Billey] for a
permutation w in S_n by downward induction on weak Bruhat order, thereby
generating the coefficients of Schubert polynomials [Lascoux-Schutzenberger]
inductively. This note provides a short and purely combinatorial proof of these
properties of mitosis.Comment: 9 pages, to appear in JCT
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
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