Equivariant Schubert calculus

Gatto, Letterio; Santiago, Taise

doi:10.1007/s11512-009-0093-5

Cited by 15 publications

(17 citation statements)

References 17 publications

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“…In this case presentation (4.10) is that the quantum equivariant cohomology ring QH * T (G(2, 4)) of the Grassmannian G (2,4) under the action of a 4-dimensional compact or algebraic torus via a diagonal action with only isolated fixed points, as studied by Mihalcea in [11,Theorem 4.2], setting p = 2 and m = 4. This is compatible with the main result of the paper [6,Theorem 3.7], with [4, Theorem 2.9] and is now a consequence of [8]. Notice that our generators are not the same as used in [11] (Cf.…”

Section: For Eachsupporting

confidence: 80%

Schubert Calculus on a Grassmann Algebra

Gatto¹,

Santiago²

2009

Can. math. bull.

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Abstract. The (classical, small quantum, equivariant) cohomology ring of the grassmannian G(k, n) is generated by certain derivations operating on an exterior algebra of a free module of rank n (Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree n into the product of two monic polynomials, one of degree k.

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Section: For Eachsupporting

confidence: 80%

Schubert Calculus on a Grassmann Algebra

Gatto¹,

Santiago²

2009

Can. math. bull.

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“…Here we will give a neat formula of the multiplication by all σ 1 p by simplifying Theorem 3.10. We remark that the classical part of our formula, i.e., the equivariant Pieri rule, is different from those known rules in [16,27]. It is obtained by simplifying Robinson's Pieri rule in a purely combinatorial way.…”

Section: By Claim C We Havementioning

confidence: 81%

On equivariant quantum Schubert calculus for G/P

Huang

2015

Journal of Algebra

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We show a Z 2 -filtered algebraic structure and a "quantum to classical" principle on the torus-equivariant quantum cohomology of a complete flag variety of general Lie type, generalizing earlier works of Leung and the second author. We also provide various applications on equivariant quantum Schubert calculus, including an equivariant quantum Pieri rule for partial flag variety F ℓ n 1 ,··· ,n k ;n+1 of Lie type A.

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“…The factors in the quotients ep 1,2,...,n ep I cancel out partially and miraculously all the summands for a fixed k turn out to be integral. For n = 3 and degree one the summands are given by the formula (21).…”

Section: Computation Of the Local Equivariant Chern Classesmentioning

confidence: 99%

Equivariant Chern Classes and Localization Theorem

Weber

2012

jsing

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For a complex variety with a torus action we propose a new method of computing Chern-Schwartz-MacPherson classes. The method does not apply resolution of singularities. It is based on the Localization Theorem in equivariant cohomology. This is an extended version of the talk given in Hefei in July 2011.Equivariant cohomology is a powerful tool for studying complex manifolds equipped with a torus action. The Localization Theorem of Atiyah and Bott and the resulting formula of Berline-Vergne allow to compute global invariants, for example invariants of singular subsets, in terms of some data attached to the fixed points of the action. We will concentrate on the equivariant Chern-Schwartz-MacPherson classes. The global class is determined by the local contributions coming from the fixed points. On the other hand, the sum of the local contributions divided by the Euler classes is equal to zero in an appropriate localization of equivariant cohomology. Especially for Grassmannians we obtain interesting formulas with nontrivial relations involving rational functions. We discuss the issue of positivity: the local equivariant Chern class may be presented in various ways, depending on the choice of generating circles of the torus. For some choices we find that the coefficients of the presentation are nonnegative. Also the coefficients in the Schur basis are nonnegative in many examples, but it turns out that not always.We begin with §1 which contains a review of the results concerning the equivariant fundamental class of an invariant subvariety in a smooth Gmanifold M . The first two chapters are valid for any algebraic group, but 1 arXiv:1110.5515v3 [math.AG]

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Equivariant Schubert calculus

Cited by 15 publications

References 17 publications

Schubert Calculus on a Grassmann Algebra

Schubert Calculus on a Grassmann Algebra

On equivariant quantum Schubert calculus for G/P

Equivariant Chern Classes and Localization Theorem

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