2,832 research outputs found
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
James reduced product schemes and double quasisymmetric functions
Symmetric function theory is a key ingredient in the Schubert calculus of
Grassmannians. Quasisymmetric functions are analogues that are similarly
central to algebraic combinatorics, but for which the associated geometry is
poorly developed. Baker and Richter (2008) showed that
manifests topologically as the cohomology ring of the loop suspension of
infinite projective space or equivalently of its combinatorial homotopy model,
the James reduced product . In recent work, we
used this viewpoint to develop topologically-motivated bases of
and initiate a Schubert calculus for in both
cohomology and -theory.
Here, we study the torus-equivariant cohomology of
. We identify a cellular basis and introduce
double monomial quasisymmetric functions as combinatorial representatives,
analogous to the factorial Schur functions and double Schubert polynomials of
classical Schubert calculus. We also provide a combinatorial
Littlewood--Richardson rule for the structure coefficients of this basis.
Furthermore, we introduce an algebro-geometric analogue of the James reduced
product construction. In particular, we prove that the James reduced product of
a complex projective variety also carries the structure of a projective
variety
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
Asymmetric function theory
The classical theory of symmetric functions has a central position in
algebraic combinatorics, bridging aspects of representation theory,
combinatorics, and enumerative geometry. More recently, this theory has been
fruitfully extended to the larger ring of quasisymmetric functions, with
corresponding applications. Here, we survey recent work extending this theory
further to general asymmetric polynomials.Comment: 36 pages, 8 figures, 1 table. Written for the proceedings of the
Schubert calculus conference in Guangzhou, Nov. 201
Skew Schubert functions and the Pieri formula for flag manifolds
We show the equivalence of the Pieri formula for flag manifolds and certain
identities among the structure constants, giving new proofs of both the Pieri
formula and of these identities. A key step is the association of a symmetric
function to a finite poset with labeled Hasse diagram satisfying a symmetry
condition. This gives a unified definition of skew Schur functions, Stanley
symmetric function, and skew Schubert functions (defined here). We also use
algebraic geometry to show the coefficient of a monomial in a Schubert
polynomial counts certain chains in the Bruhat order, obtaining a new
combinatorial construction of Schubert polynomials.Comment: 24 pages, LaTeX 2e, with epsf.st
Billey's formula in combinatorics, geometry, and topology
In this expository paper we describe a powerful combinatorial formula and its
implications in geometry, topology, and algebra. This formula first appeared in
the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey
discovered it independently five years later, and it played a prominent role in
her work to evaluate certain polynomials closely related to Schubert
polynomials.
Billey's formula relates many pieces of Schubert calculus: the geometry of
Schubert varieties, the action of the torus on the flag variety, combinatorial
data about permutations, the cohomology of the flag variety and of the Schubert
varieties, and the combinatorics of root systems (generalizing inversions of a
permutation). Combinatorially, Billey's formula describes an invariant of pairs
of elements of a Weyl group. On its face, this formula is a combination of
roots built from subwords of a fixed word. As we will see, it has deeper
geometric and topological meaning as well: (1) It tells us about the tangent
spaces at each permutation flag in each Schubert variety. (2) It tells us about
singular points in Schubert varieties. (3) It tells us about the values of
Kostant polynomials. Billey's formula also reflects an aspect of GKM theory,
which is a way of describing the torus-equivariant cohomology of a variety just
from information about the torus-fixed points in the variety.
This paper will also describe some applications of Billey's formula,
including concrete combinatorial descriptions of Billey's formula in special
cases, and ways to bootstrap Billey's formula to describe the equivariant
cohomology of subvarieties of the flag variety to which GKM theory does not
apply.Comment: 14 pages, presented at the International Summer School and Workshop
on Schubert Calculus in Osaka, Japan, 201
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