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| Quiver polynomials and Schubert varieties |
| Abstract: Given a sequence E_0 --> ... --> E_n
of maps between vector bundles over a scheme, and integers r_ij for i <
j, one can ask what is the cohomology class of the locus where the rank
of the composite map
E_i --> E_j is bounded above by r_ij for all i < j. This class turns out to be a polynomial, called the quiver polynomial, in the Chern roots of the vector bundles E_i. Quiver polynomials can alternativley be characterized as equivariant cohomology classes of orbit closures in the space of quiver representations, under the action of the general linear group that changes bases. These orbit closures are closely related to certain Schubert varieties in partial flag manifolds. Exploiting this relationship produces a number of positive combinatorial formulae for quiver polynomials. This talk describes joint work with Allen Knutson and Mark Shimozono. |