Residue formulas for push-forwards in equivariant cohomology: a symplectic approach

Zielenkiewicz, Magdalena

doi:10.4310/jsg.2018.v16.n5.a7

Cited by 3 publications

(4 citation statements)

References 12 publications

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“…They were generalized to Grassmannians for the classical groups in [24]. A conceptual proof based on the Jeffrey-Kirwan theory is presented in [25]. Similar formulas can be obtained in equivariant K-theory.…”

Section: Introductionmentioning

confidence: 94%

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Residues formulas for the push-forward in K-theory, the case of $$\mathbf{G}_2/P$$

Weber

Zielenkiewicz

2018

J Algebr Comb

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We study residue formulas for push-forward in the equivariant K-theory of homogeneous spaces. For the classical Grassmannian, the residue formula can be obtained from the cohomological formula by a substitution. We also give another proof using symplectic reduction and the method involving the localization theorem of Jeffrey-Kirwan. We review formulas for other classical groups, and we derive them from the formulas for the classical Grassmannian. Next, we consider the homogeneous spaces for the exceptional group G 2. One of them, G 2 /P 2 corresponding to the shorter root, embeds in the Grassmannian Gr(2, 7). We find its fundamental class in the equivariant K-theory K T (Gr(2, 7)). This allows to derive a residue formula for the push-forward. It has significantly different character compared to the classical groups case. The factor involving the fundamental class of G 2 /P 2 depends on the equivariant variables. To perform computations more efficiently, we apply the basis of K-theory consisting of Grothendieck polynomials. The residue formula for push-forward remains valid for the homogeneous space G 2 /B as well.

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Section: Introductionmentioning

confidence: 94%

“…The formula (4) can be deduced from the cohomological push-forward formula (see [22], Chapter 4. or [24], Corollary 3.2) or can be proved directly using symplectic reduction as in [25,26].…”

Section: Proof Of Residue Formulasmentioning

confidence: 99%

Residues formulas for the push-forward in K-theory, the case of $$\mathbf{G}_2/P$$

Weber

Zielenkiewicz

2018

J Algebr Comb

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“…They were generalized to Grassmannians for the classical groups in [Zie14]. A conceptual proof based on the Jeffrey-Kirwan theory is presented in [Zie16]. Similar formulas can be obtained in equivariant K-theory.…”

Section: Introductionmentioning

confidence: 94%

“…The formula (4) can be deduced from the cohomological push-forward formula (see [Web12], Chapter 4. or [Zie14], Corollary 3.2) or can be proved directly using symplectic reduction as in [Zie16,Zie17].…”

Section: Proof Of Residue Formulasmentioning

confidence: 99%

Residues formulas for the push-forward in K-theory, the case of G2/P

Weber¹,

Zielenkiewicz²

2017

Preprint

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We study residue formulas for push-forward in the equivariant K-theory of homogeneous spaces. For the classical Grassmannian the residue formula can be obtained from the cohomological formula by a substitution. We also give another proof using symplectic reduction and the method involving the localization theorem of Jeffrey-Kirwan. We review formulas for other classical groups, and we derive them from the formulas for the classical Grassmannian. Next we consider the homogeneous spaces for the exceptional group G2. One of them, G2/P2 corresponding to the shorter root, embeds in the Grassmannian Gr(2, 7). We find its fundamental class in the equivariant K-theory K T (Gr(2, 7)). This allows to derive a residue formula for the push-forward. It has significantly different character comparing to the classical groups case. The factor involving the fundamental class of G2/P2 depends on the equivariant variables. To perform computations more efficiently we apply the basis of K-theory consisting of Grothendieck polynomials. The residue formula for push-forward remains valid for the homogeneous space G2/B as well.

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Integrable models and K-theoretic pushforward of Grothendieck classes

Motegi

2021

Nuclear Physics B

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Residue formulas for push-forwards in equivariant cohomology: a symplectic approach

Cited by 3 publications

References 12 publications

Residues formulas for the push-forward in K-theory, the case of $$\mathbf{G}_2/P$$

Residues formulas for the push-forward in K-theory, the case of $$\mathbf{G}_2/P$$

Residues formulas for the push-forward in K-theory, the case of G2/P

Integrable models and K-theoretic pushforward of Grothendieck classes

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