TY - JOUR

T1 - New techniques for obtaining schubert-type formulas for hamiltonian manifolds

AU - Sabatini, Silvia

AU - Tolman, Susan

PY - 2013

Y1 - 2013

N2 - In [GT], Goldin and Tolman extend some ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. (See also [Kn99,Kn08].) The main goal of this paper is to build on this work by finding more effective formulas. More explicitly, given a generic component of the moment map, they define a canonical class αp in the equivariant cohomology of the manifold M for each fixed point p ∈ M. When they exist, canonical classes form a natural basis of the equivari-ant cohomology of M. In particular, when M is a flag variety, these classes are the equivariant Schubert classes. It is a long-standing problem in combinatorics to find positive integral formulas for the equivariant structure constants associated to this basis. Since computing the restriction of the canonical classes to the fixed points determines these structure constants, it is important to find effective formulas for these restrictions. In this paper, we introduce new techniques for calculating the restrictions of a canonical class αp to a fixed point q. Our formulas are nearly always simpler, in the sense that they count the contributions over fewer paths. Moreover, our formula is manifestly positive and integral in certain important special cases.

AB - In [GT], Goldin and Tolman extend some ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. (See also [Kn99,Kn08].) The main goal of this paper is to build on this work by finding more effective formulas. More explicitly, given a generic component of the moment map, they define a canonical class αp in the equivariant cohomology of the manifold M for each fixed point p ∈ M. When they exist, canonical classes form a natural basis of the equivari-ant cohomology of M. In particular, when M is a flag variety, these classes are the equivariant Schubert classes. It is a long-standing problem in combinatorics to find positive integral formulas for the equivariant structure constants associated to this basis. Since computing the restriction of the canonical classes to the fixed points determines these structure constants, it is important to find effective formulas for these restrictions. In this paper, we introduce new techniques for calculating the restrictions of a canonical class αp to a fixed point q. Our formulas are nearly always simpler, in the sense that they count the contributions over fewer paths. Moreover, our formula is manifestly positive and integral in certain important special cases.

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U2 - 10.4310/JSG.2013.v11.n2.a3

DO - 10.4310/JSG.2013.v11.n2.a3

M3 - Article

AN - SCOPUS:84876214602

SN - 1527-5256

VL - 11

SP - 179

EP - 230

JO - Journal of Symplectic Geometry

JF - Journal of Symplectic Geometry

IS - 2

ER -