Wall-crossings and a categorification of <i>K</i>-theory stable bases of the Springer resolution

Su, Changjian; Zhao, Gufang; Zhong, Changlong

doi:10.1112/s0010437x21007533

Cited by 4 publications

(2 citation statements)

References 35 publications

(24 reference statements)

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“…All the results stated here are known, but we provide proofs in the new setup (using the '⊙'-action), which give new insight and are sometimes simpler. Moreover, the basis in Deodhar's modules discussed in §4 is closely related to the stable basis in the K-theory of the Springer resolution defined by Maulik and Okounkov [31]; an investigation of the latter is contained in [37].…”

Section: All Parabolic Bott-samelson Classes ζ Jmentioning

confidence: 99%

Parabolic Kazhdan–lusztig Basis, Schubert Classes, and Equivariant Oriented Cohomology

Lenart

Zainoulline²,

Zhong

2019

J. Inst. Math. Jussieu

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We study the equivariant oriented cohomology ring hT (G/P ) of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott-Samelson classes in hT (G/P ) can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results by Brion, Knutson, Peterson, Tymoczko and others.We then focus on the equivariant oriented cohomology theory corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar's construction of the parabolic Kazhdan-Lusztig basis. Based on it, we define the parabolic Kazhdan-Lusztig (KL) Schubert classes independently of a reduced word. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We then prove several special cases. 3.6. An example 13 3.7. A generalized Deodhar's resolution 15 4. Deodhar's parabolic Hecke modules revisited 17 4.1. The case of a multiplicative formal group law 17 4.2. Modules over the Hecke algebra

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Section: All Parabolic Bott-samelson Classes ζ Jmentioning

confidence: 99%

Parabolic Kazhdan–lusztig Basis, Schubert Classes, and Equivariant Oriented Cohomology

Lenart

Zainoulline²,

Zhong

2019

J. Inst. Math. Jussieu

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“…Homogenous varieties

G/P

are our main examples of varieties satisfying the local product condition. The study of characteristic classes and stable envelopes of such varieties is an important theme present in recent research (for example, [2, 37, 40, 44]). A priori the stable envelope is defined for symplectic varieties which admit a proper map to an affine variety.…”

Section: Introductionmentioning

confidence: 99%

Comparison of motivic Chern classes and stable envelopes for cotangent bundles

Jakub

2022

Journal of Topology

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Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes

Koncki,

Weber

2024

Math. Ann.

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Let G be a linear semisimple algebraic group and B its Borel subgroup. Let $${\mathbb {T}}\subset B$$ T ⊂ B be the maximal torus. We study the inductive construction of Bott–Samelson varieties to obtain recursive formulas for the twisted motivic Chern classes of Schubert cells in G/B. To this end we introduce two families of operators acting on the equivariant K-theory $${\text {K}}_{\mathbb {T}}(G/B)[y]$$ K T ( G / B ) [ y ] , the right and left Demazure–Lusztig operators depending on a parameter. The twisted motivic Chern classes coincide (up to normalization) with the K-theoretic stable envelopes. Our results imply wall-crossing formulas for a change of the weight chamber and slope parameters. The right and left operators generate a twisted double Hecke algebra. We show that in the type A this algebra acts on the Laurent polynomials. This action is a natural lift of the action on $${\text {K}}_{\mathbb {T}}(G/B)[y]$$ K T ( G / B ) [ y ] with respect to the Kirwan map. We show that the left and right twisted Demazure–Lusztig operators provide a recursion for twisted motivic Chern classes of matrix Schubert varieties.

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Wall-crossings and a categorification of K-theory stable bases of the Springer resolution

Cited by 4 publications

References 35 publications

Parabolic Kazhdan–lusztig Basis, Schubert Classes, and Equivariant Oriented Cohomology

Parabolic Kazhdan–lusztig Basis, Schubert Classes, and Equivariant Oriented Cohomology

Comparison of motivic Chern classes and stable envelopes for cotangent bundles

Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes

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