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Abstracts for AMS Special SessionRing Structures in the Schubert Calculus
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We present a geometric proof of the Horn Conjecture and obtain the
Saturation conjecture for Gl(n) as a consequence.
The Horn conjecture says
that non vanishing of Littlewood-Richardson numbers
clm,n is
equivalent to the semi-stability of a certain N-filtered vector space (for
generic flags). Knutson-Tao gave combinatorial proofs of the Saturation
conjecture using the theory of Honeycombs. Fulton showed that Saturation implies
the Horn Conjecture. Our proof here is independent of these previous works and
raises interesting questions on the Quantum analogues of the Horn Conjecture.
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Soergel has recently shown that knowledge of torsion in the intersection
cohomology of Schubert varieties leads to information about modular
representation theory. In particular, he has asked what is the smallest
N > 0 for which the decomposition theorem holds with coefficients in
Z[1/N] for a given Bott-Samelson resolution
X --> Xw of a Schubert variety Xw.
This implies that there is no
Z/pZ-torsion in the intersection cohomology stalks of
Xw for p|N.
We describe certain polynomials representing a basis for the equivariant
cohomology H*T(X) which can be used
to calculate N.
We present examples in types A7 and
D4 for which N > 1.
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Universal Schubert polynomials, which give the answer to
a degeneracy locus problem, specialize to quantum and ordinary
Schubert polynomials. I present a Pieri-type formula in some special
cases.
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We reframe the question of equivariant Schubert calculus of complete
flag varieties in terms of Bott-Samelson manifolds, desingularizations
of the flag varieties. In particular, there is a more general algebra
related to the equivariant cohomology ring of a Bott-Samelson sitting
over a Schubert variety, and we show how to multiply in this algebra.
One result is a non-positive formula (and geometric interpretation
thereof) for Schubert calculus. The same formula was discovered
combinatorially by Sara Billey. Although this is not a positive
calculus, there are restricted cases in which it gives a positive
formula. Perhaps more interestingly, it sheds light on some geometry
behind Schubert calculus. The work we present is joint with Allen
Knutson. Some of it was independently discovered by Mathieu Willems
(and can be found on the math archives).
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The action of a complex torus on a non-singular projective
variety is a GKM action if there are finitely many fixed points and
finitely many one-dimensional orbits. For such an action the fixed points
and one-dimensional orbits can be viewed as the vertices and edges of a
regular graph, and Goresky, Kottwitz and MacPherson showed that the
equivariant cohomology ring of the space can be read off from this graph.
In this talk we show that an analogous result is true for "GKM spaces
with non-isolated fixed points". The work I will report on is joint with
Tara Holm.
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Quantum Schubert calculus exhibits a number of symmetries which lead to
equalities of certain Gromov Witten invariants, mainly a
Z/n action and
an involution related to complex conjugation. We analyze the structure
of these symmetries and give applications to quantum Schubert calculus.
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We derive a recurrence on Schubert structure constants from the noncomplex Weyl group action on G/B. It is occasionally positive; for example it computes
We construct a flat degeneration of the flag manifold to the
toric variety Y associated to the Gel'fand-Cetlin polytope. Every
Schubert variety Xw degenerates to a reduced union of toric
subvarieties of Y, generalizing results of Gonciulea and Lakshmibai. The
faces of the Gel'fand-Cetlin polytope corresponding to the components of
the degeneration of Xw are precisely those given by rc-graphs of
Fomin-Kirillov. This is joint work with Ezra Miller.
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We present some explicit multiplication formulas in the Grothendieck
ring of the variety of complete flags in Cn, which
generalize some well-known formulas in cohomology. Our formulas involve
Grothendieck polynomials, which are representatives for Schubert classes
in the Grothendieck ring. These formulas are concerned with: (1) the
hyperplane section of a Schubert variety (a Monk type formula); (2) the
restriction of a dominant line bundle to a Schubert variety; (3) the
multiplication of an arbitrary Schubert class by a special one, which is
pulled back from a Grassmannian projection (a Pieri type formula). We
show the way in which formulas (1) and (2) are related by convolution
inverse in a certain Hopf algebra. Different and more general formulas
for (1) and (2) were given by A. Ram, H. Pittie, and M. Brion. Their
formulas are in terms of Littelmann paths, and are only shown to hold in
the Grothendieck ring. Our formulas are polynomial identities, and are
in terms of chains in the Bruhat order on the symmetric group. The
latter framework seems more suitable for multiplication problems in
Schubert calculus, as shown by previous work in this area. The part of
this work related to problem (3) is joint with F. Sottile.
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A theorem of Kim gives a presentation of the (small)
quantum cohomology ring of the generalized flag manifold
G/B. In order to have a complete picture of this
ring, one needs a solution to the "quantum Giambelli
problem", i.e. polynomial representatives of Schubert
classes in Kim's ring. We present a general formula
for such polynomials, which involves representatives
of Schubert classes in Borel's description of
H*(G/B),
divided difference operators and the coefficients of the
"quantum Chevalley formula".
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Fix vector spaces V0, ..., Vn,
and consider the set Hom(V) of quivers
V0 --> ... --> Vn.
Quiver cycles are subsets
Or of Hom(V) where the ranks of
the composite maps Vi --> Vj
are bounded above by specified integers
r = rij for i < j.
Buch and Fulton expressed the equivariant
cohomology class [Or] in terms of Schur functions,
and conjectured a
combinatorial formula for the coefficients. In particular, they
conjectured that the coefficients, which directly generalize
Littlewood-Richardson coefficients, are positive.
In this ongoing project, we construct a flat family whose general fiber
is isomorphic to Or, and whose special fiber
has components that are
products of matrix Schubert varieties.
This proves a formula for [Or]
in terms of Stanley symmetric functions (stable double Schubert
polynomials)
Fw indexed by lists w of n permutations. Our
formula is obviously positive for geometric reasons, so it immediately
implies positivity of the Buch-Fulton formula. Moreover, we conjecture
that the special fiber is generically reduced, so that our coefficients
all equal 1, and we propose a simple nonrecursive combinatorial
characterization of which lists w appear.
This is joint work with Allen Knutson and Mark Shimozono.
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The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of G/B. We enhance a result of Fulton and Woodward by showing that the minimal monomial in the quantum parameters that occurs in the quantum product of two Schubert classes has a simple interpretation in terms of directed paths in this graph.
We define path Schubert polynomials, which are quantum cohomology analogues
of skew Schubert polynomials recently introduced by Lenart and Sottile.
They are given by sums over paths in the quantum Bruhat graph of type A.
The 3-point Gromov-Witten invariants for the flag manifold are expressed
in terms of these polynomials. This construction gives a combinatorial
description for the set of all monomials in the quantum parameters that
occur in the quantum product of two Schubert classes.
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For any complex semisimple lie group G, we know that in the ordinary
cohomology ring H*(G/B; Z), many of
the structure constants (for the Schubert basis) vanish, and the rest are
strictly positive.
We present a combinatorial game which provides some criteria for determining
which of these Schubert structure constants vanish.
Although these criteria are not proven to cover all cases, in practice
they work very well, giving a complete answer to the question for
G = GL(7,C).
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We discuss an approach to Schubert Calculus based on a theory of
quasideterminats of matrices with noncommutative entries developed by I. Gelfand
and W. Retakh.
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The Peterson variety is a projective variety
introduced by Dale Peterson that encodes the quantum cohomology
rings of all partial flag varieties as coordinate rings coming
from an affine paving. We will describe its `positive part'
(in the sense of total positivity for matrices). This is
related to Schubert bases and the positivity of structure
constants in quantum Schubert calculus.
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In the lecture we approach the theory of Schur and Schubert
polynomials from the point of view of generalized Thom polynomials,
i.e. we realize these polynomials as equivariant Poincare duals of
orbits of some representation. Then we can apply the author's method
to compute them as Thom polynomials, thus we get new definitions for
Schur and Schubert polynomials. Interestingly enough we will obtain
them as unique solutions of linear equation systems. Other definitions
for Schur and Schubert polynomials (e.g. the Lascoux-Schützenberger
recursion) will be interpreted in this language. Along the way we also
redefine the double Schubert polynomials and define "double Schur
polynomials". We show how this approach is related to the structure
of the cohomology ring of the Grassmannian, the flag manifold and
other quotient spaces.
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From geometric considerations we construct Grothendieck
polynomials in terms of chains in the Bruhat order on the symmetric
group. We also give formulas for the specialization of a Grothendieck
polynomial at two sets of variables in terms of Grothendieck polynomials
in each set of variables, permutation patterns, and k-theory structure
constants of flag manifolds.
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We discuss a number of results which show that the 3-point,
genus zero, degree d Gromov-Witten invariants on a Grassmannian
manifold X are equal to classical triple intersection numbers on a
homogeneous space Yd of the same Lie type as X. In the symplectic
and orthogonal cases, X is the Grassmannian which parametrizes maximal
isotropic subspaces of a vector space equipped with the corresponding
nondegenerate bilinear form. Our theorems are proved using only basic
algebraic geometry and can be applied to study the small quantum
cohomology ring of X. This is joint work with Anders S. Buch and
Andrew Kresch.
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In an attempt to provide an alternative concrete framework for studying
the Schubert calculus of G/B, we describe a general construction for
bases of the cohomology ring of G/B in terms of "special classes".
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In the GKM picture of the equivariant cohomology of flag varieties, a Schubert class Sw is regarded as a map (satisfying some compatibility conditions) that attaches a polynomial to each fixed point. The equivariant Schubert class Sw can be computed by applying a left divided difference operator (associated to a certain reduced word) to the top class, and the value Sw(u) is obtained by adding the (rational) contributions of certain subwords. For Grassmannians, these subwords correspond bijectively to fixed points for a torus action on a smooth manifold, and the computation of Sw(u) is equivalent to integrating an equivariant form over this manifold.