A GKM description of the equivariant cohomology ring of a homogeneous space

Guillemin, Victor; Holm, Tara S.; Zara, Catalin

doi:10.1007/s10801-006-6027-4

Cited by 65 publications

(92 citation statements)

References 9 publications

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“…One of our main applications of this result will be a fiber bundle version of the main result in [4]. In [4], it is shown that if G is a compact semisimple Lie group, T a Cartan subgroup, and K a closed subgroup of G, then the following conditions 3890 V. Guillemin et al are equivalent:…”

Section: Theorem 13 a Functionmentioning

confidence: 97%

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Balanced Fiber Bundles and GKM Theory

Guillemin

Sabatini

Zara

2012

International Mathematics Research Notices

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Let T be a torus and B a compact T −manifold. Goresky, Kottwitz, and MacPherson show in [GKM] that if B is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant cohomology ring H * T (B) as a subring of H * T (B T ). In this paper we prove an analogue of this result for T −equivariant fiber bundles: we show that if M is a T −manifold and π : M → B a fiber bundle for which π intertwines the two T −actions, there is a simple combinatorial description of H * T (M ) as a subring of H * T (π −1 (B T )). Using this result we obtain fiber bundle analogues of results of [GHZ] on GKM theory for homogeneous spaces.

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Section: Theorem 13 a Functionmentioning

confidence: 97%

“…In [4], it is shown that if G is a compact semisimple Lie group, T a Cartan subgroup, and K a closed subgroup of G, then the following conditions 3890 V. Guillemin et al are equivalent:…”

Section: Theorem 13 a Functionmentioning

confidence: 99%

Balanced Fiber Bundles and GKM Theory

Guillemin

Sabatini

Zara

2012

International Mathematics Research Notices

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“…We note that the structure of Γ has many common features with its classical analogue (see [8,22,27]). …”

Section: Theorem 11 the Number Of One-dimensional T -Orbits In F Amentioning

confidence: 99%

“…The 22 smooth torus . 3 The moment graph around vertex (22) fixed points are highlighted by a frame. These are the vertices adjacent to precisely 6 = dim F a 4 edges.…”

Section: Appendix Cmentioning

confidence: 99%

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Degenerate flag varieties: moment graphs and Schröder numbers

2012

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We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T . Using the group action, we describe the moment graphs, encoding the zero-and one-dimensional T -orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schröder number and the Poincaré polynomial is given by a natural statistics counting the number of diagonal steps in a Schröder path. As an application we obtain a new combinatorial description of the large and small Schröder numbers and their q-analogues.

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GKM Graph Locally Modeled by $$T^{n}\times S^{1}$$-Action on $$T^{*}\mathbb {C}^{n}$$ and Its Graph Equivariant Cohomology

Kuroki,

Uma

2024

Fields Institute Communications

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A GKM description of the equivariant cohomology ring of a homogeneous space

Cited by 65 publications

References 9 publications

Balanced Fiber Bundles and GKM Theory

Balanced Fiber Bundles and GKM Theory

Degenerate flag varieties: moment graphs and Schröder numbers

GKM Graph Locally Modeled by $$T^{n}\times S^{1}$$-Action on $$T^{*}\mathbb {C}^{n}$$ and Its Graph Equivariant Cohomology

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