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Intersection Cohomology of Rank One Local Systems for Arrangement Schubert Varieties
In this thesis we study the intersection cohomology of arrangement Schubert varieties with coefficients in a rank one local system on a hyperplane arrangement complement. We prove that the intersection cohomology can be computed recursively in terms of certain polynomials, if a local system has only monodromies. In the case where the hyperplane arrangement is generic central or equivalently the associated matroid is uniform and the local system has only monodromies, we prove that the intersection cohomology is a combinatorial invariant. In particular when the hyperplane arrangement is associated to the uniform matroid of rank over elements, and the local system has monodromies, we can give a closed formula for the intersection cohomology
Schubert polynomials and Arakelov theory of symplectic flag varieties
Let X be the flag variety of the symplectic group. We propose a theory of
combinatorially explicit Schubert polynomials which represent the Schubert
classes in the Borel presentation of the cohomology ring of X. We use these
polynomials to describe the arithmetic Schubert calculus on X. Moreover, we
give a method to compute the natural arithmetic Chern numbers on X, and show
that they are all rational numbers.Comment: 22 pages; final versio
Schubert calculus and shifting of interval positroid varieties
Consider k x n matrices with rank conditions placed on intervals of columns.
The ranks that are actually achievable correspond naturally to upper triangular
partial permutation matrices, and we call the corresponding subvarieties of
Gr(k,n) the _interval positroid varieties_, as this class lies within the class
of positroid varieties studied in [Knutson-Lam-Speyer]. It includes Schubert
and opposite Schubert varieties, and their intersections, and is Grassmann dual
to the projection varieties of [Billey-Coskun].
Vakil's "geometric Littlewood-Richardson rule" [Vakil] uses certain
degenerations to positively compute the H^*-classes of Richardson varieties,
each summand recorded as a (2+1)-dimensional "checker game". We use his same
degenerations to positively compute the K_T-classes of interval positroid
varieties, each summand recorded more succinctly as a 2-dimensional "K-IP pipe
dream". In Vakil's restricted situation these IP pipe dreams biject very simply
to the puzzles of [Knutson-Tao].
We relate Vakil's degenerations to Erd\H os-Ko-Rado shifting, and include
results about computing "geometric shifts" of general T-invariant subvarieties
of Grassmannians.Comment: 35 pp; this subsumes and obviates the unpublished
http://arxiv.org/abs/1008.430
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