On the quantum product of Schubert classes

Fulton, William; Woodward, Chris

doi:10.1090/s1056-3911-04-00365-0

Cited by 124 publications

(174 citation statements)

References 32 publications

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“…We have not been able to turn this into a combinatorial characterization of the smallest power, as Fulton and Woodward did ( [11], Proposition 9.1(2)) in the case s = 2. It should be noted that Theorem 1.3 gives no information on the other powers of q that appear in the same product.…”

Section: Quantum Product Of Schubert Classesmentioning

confidence: 90%

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Quantum generalization of the Horn conjecture

Belkale

2007

J. Amer. Math. Soc.

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Section: Quantum Product Of Schubert Classesmentioning

confidence: 90%

“…The case of two factors for an arbitrary G/P was considered by Fulton and Woodward in [11]. Recall that (by degree considerations), if…”

Section: Quantum Product Of Schubert Classesmentioning

confidence: 99%

Quantum generalization of the Horn conjecture

Belkale

2007

J. Amer. Math. Soc.

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“…The integer d min (ν ∨ , μ) is the minimal power appearing in the quantum product s μ * s ν ; see [16]. In fact, there exists an interval of integers…”

Section: Lattice Configurations and Quantum Kostka Numbersmentioning

confidence: 99%

Quantum Cohomology via Vicious and Osculating Walkers

Korff

2014

Lett Math Phys

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Abstract.We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang-Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gaugedû(n) k -WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further that the partition functions of the vicious and osculating walker model are given in terms of Postnikov's toric Schur functions and can be interpreted as generating functions for GromovWitten invariants. We reveal an underlying quantum group structure in terms of YangBaxter algebras and use it to give a generating formula for toric Schur functions in terms of divided difference operators which appear in known representations of the nil-Hecke algebra.Mathematics Subject Classification (2010). 14N35, 05E05, 05A15, 05A19, 82B23.

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“…The basic line bundle over RK (X; /-tl, ---, /-Lb) determines a projective embedding with Hilbert polynomial given by the dimension of the space of genus zero conformal blocks A~/~1,..., at level k with markings knpb, [41], [35], [51] [21], [22]. These GromovWitten invariants of G/P (as opposed to the invariants that appear in the large quantum cohomology) are computable in practice using formulas of D. Peterson [42], whose proofs are given in [22] and [54]. An Let Vectr denote the functor which assigns to any complex manifold S, the isomorphism classes of r-equivariant bundles E -X.…”

mentioning

confidence: 99%

Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants

Teleman¹,

Woodward²

2003

Annales De L’institut Fourier

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The set of conjugacy classes appearing in a product of conjugacy classes in a compact, 1-connected Lie group K can be identified with a convex polytope in the Weyl alcove [37]. In this paper we identify linear inequalities defining this polytope. Each inequality corresponds to a non-vanishing Gromov-Witten invariant for a generalized flag variety G/P , where G is the complexification of K and P is a maximal parabolic subgroup. This generalizes the results for SU (n) of Agnihotri and the second author [1] and Belkale [7] on the eigenvalues of a product of unitary matrices and quantum cohomology of Grassmannians.

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On the quantum product of Schubert classes

Abstract: Abstract. We give a formula for the smallest powers of the quantum parameters q that occur in a product of Schubert classes in the (small) quantum cohomology of general flag varieties G/P . We also include a complete proof of Peterson's quantum

Cited by 124 publications

References 32 publications

Quantum generalization of the Horn conjecture

Quantum generalization of the Horn conjecture

Quantum Cohomology via Vicious and Osculating Walkers

Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants

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