19 research outputs found
K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles splitting as direct sums
We use equivariant localization and divided difference operators to determine
formulas for the torus-equivariant fundamental cohomology classes of -orbit
closures on the flag variety for various symmetric pairs . In type
, we realize the closures of K=GL(p,\C) \times GL(q,\C)-orbits on
GL(p+q,\C)/B as universal degeneracy loci for a vector bundle over a variety
which is equipped with a single flag of subbundles and which splits as a direct
sum of subbundles of ranks and . The precise description of such a
degeneracy locus relies upon knowing a set-theoretic description of -orbit
closures, which we provide via a detailed combinatorial analysis of the poset
of "-clans," which parametrize the orbit closures. We describe precisely
how our formulas for the equivariant classes of -orbit closures can be
interpreted as formulas for the classes of such degeneracy loci in the Chern
classes of the bundles involved. In the cases outside of type , we suggest
that the orbit closures should parametrize degeneracy loci involving a vector
bundle equipped with a non-degenerate symmetric or skew-symmetric bilinear
form, a single flag of subbundles which are isotropic or Lagrangian with
respect to the form, and a splitting as a direct sum of subbundles with each
summand satisfying some property (depending on ) with respect to the form.
The precise description of such a degeneracy locus is conjectured for all cases
in types and .Comment: Final version, to appear in Geom. Dedicat
GL(p) x GL(q)-orbit closures on the flag variety and Schubert structure constants for (p,q)-pairs
We give positive combinatorial descriptions of Schubert structure constants
for the full flag variety in type when and form
what we refer to as a "-pair" (). The key observation is that a
certain subset of the -orbit closures
on the flag variety (those satisfying an easily stated pattern avoidance
condition) are Richardson varieties. The result on structure constants follows
when one combines this observation with a theorem of Brion concerning
intersection numbers of spherical subgroup orbit closures and Schubert
varieties.Comment: This paper is now replaced by arXiv:1209.0739. I am leaving this here
only because the published version of arXiv:1109.2574 refers to this version,
and specific results therei
K-orbit closures and Barbasch-Evens-Magyar varieties
We define the Barbasch-Evens-Magyar variety. We show it is isomorphic to the
smooth variety defined in [D. Barbasch-S. Evens '94] that maps finite-to-one to
a symmetric orbit closure, thereby giving a resolution of singularities in
certain cases. Our definition parallels [P. Magyar '98]'s construction of the
Bott-Samelson variety [H. C. Hansen '73, M. Demazure '74]. From this
alternative viewpoint, one deduces a graphical description in type A,
stratification into closed subvarieties of the same kind, and determination of
the torus-fixed points. Moreover, we explain how these manifolds inherit a
natural symplectic structure with Hamiltonian torus action. We then prove that
the moment polytope is expressed in terms of the moment polytope of a
Bott-Samelson variety.Comment: 26 pages, 4 figure
Schubert calculus of Richardson varieties stable under spherical Levi subgroups
We observe that the expansion in the basis of Schubert cycles for
of the class of a Richardson variety stable under a spherical Levi subgroup is
described by a theorem of Brion. Using this observation, along with a
combinatorial model of the poset of certain symmetric subgroup orbit closures,
we give positive combinatorial descriptions of certain Schubert structure
constants on the full flag variety in type . Namely, we describe
when and are inverse to Grassmannian permutations with unique descents
at and , respectively. We offer some conjectures for similar rules in
types and , associated to Richardson varieties stable under spherical
Levi subgroups of SO(2n+1,\C) and SO(2n,\C), respectively.Comment: Section 4 significantly shortened, and other minor changes made as
suggested by referees. Final version, to appear in Journal of Algebraic
Combinatoric
K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form
We use equivariant localization and divided difference operators to determine
formulas for the torus-equivariant fundamental cohomology classes of -orbit
closures on the flag variety , where G = GL(n,\C), and where is one
of the symmetric subgroups O(n,\C) or Sp(n,\C). We realize these orbit
closures as universal degeneracy loci for a vector bundle over a variety
equipped with a single flag of subbundles and a nondegenerate symmetric or
skew-symmetric bilinear form taking values in the trivial bundle. We describe
how our equivariant formulas can be interpreted as giving formulas for the
classes of such loci in terms of the Chern classes of the various bundles.Comment: Minor revisions and corrections suggested by referees. Final version,
to appear in Transformation Group