Twisted motivic Chern class and stable envelopes

Jakub, Koncki,; Weber, Andrzej

doi:10.1016/j.aim.2022.108374

Cited by 3 publications

(5 citation statements)

References 21 publications

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“…It is proved in [KW22,corollary 4.6] that this class is well defined, i.e. it does not depend on a choice of a SNC resolution of singularities.…”

Section: Functoriality: For An Equivariant Proper Mapmentioning

confidence: 99%

“…Twisted class. The twisted motivic Chern class was defined in [KW22]. We repeat its definition here.…”

Section: Functoriality: For An Equivariant Proper Mapmentioning

confidence: 99%

“…The equivariant versions are due to [Ohm06,Web16,FRW21,AMSS19]. In [KW22] the twisted version of the motivic Chern class was defined. It takes into account a chosen fractional line bundle.…”

Section: Introductionmentioning

confidence: 99%

“…In the K-theory, a comparison between the stable envelope for a small anti-ample slope and the motivic Chern class was carried out in [AMSS19,FRW21,Kon22]. The twisted class of [KW22] is a generalization of the motivic Chern class which agrees with the stable envelope for a general enough slope. It allows to define the stable envelope in terms of a resolution of singularities of the Schubert variety.…”

Section: Introductionmentioning

confidence: 99%

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

Jakub¹,

Weber²

2023

Preprint

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Let G be a linear semisimple algebraic group and B its Borel subgroup. Let 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 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 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 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.Both authors supported by NCN grant 2016/23/G/ST1/04282 (Beethoven 2). 3.3. Reflections 4. Bott-Samelson resolution 4.1. Inductive construction 4.2. Pull-backs of line bundles 5. Twisted motivic Chern class 5.1. Motivic Chern class 5.2. Twisted class 5.3. Twisted classes of Schubert cells and stable envelopes 6. Twisted Demazure-Lusztig operators 6.1. The definitions and main properties 6.2. Right Demazure-Lusztig operators 6.3. Bott-Samelson recursion 7. Twisted Hecke algebra 8. Left induction 8.1. Recursive formula for G/B 8.2. Recursion for G/P 8.3. Wall crossing for change of the coweight chamber 9. Wall-crossing formula for a change of the slope 10. Upgrade from GL n /B to End(C n ). 10.1. The Kirwan map 10.2. Matrix Schubert varieties 10.3. The lift of boundary divisors 11. Universal algebra 11.1. The small algebra 11.2. Operators with parameters in t * n ⊂ gl * n 11.3. Comparison with K T (GL n /B n ) 12. Comparison with the matrix Schubert varieties 12.1. Corollaries from the work on square-zero matrices 12.2. Twisted motivic Chern classes of matrix Schubert varieties 13. Resolution of matrix Schubert varieties 13.1. The left resolution 13.2. The right resolution 14. Proof of theorem 12.6 14.1. Left induction 14.2. Right induction 15. Twisted double Hecke algebra of the general type 15.1. Type C 2 15.2. Type G 2 References NotationsAll considered varieties are complex and quasi-projective. We consider an algebraic torus T ≃ (C * ) rk T .2.1. Line bundles and divisors. Let X be a quasiprojective T-variety and L → X an equivariant line bundle i.e. a line bundle together with a linearization. For any fixed point x ∈ X T , the fiber L |x is a representation of the torus T.

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“…It is proved in [KW22,corollary 4.6] that this class is well defined, i.e. it does not depend on a choice of a SNC resolution of singularities.…”

Section: Functoriality: For An Equivariant Proper Mapmentioning

confidence: 99%

“…Twisted class. The twisted motivic Chern class was defined in [KW22]. We repeat its definition here.…”

Section: Functoriality: For An Equivariant Proper Mapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Jakub¹,

Weber²

2023

Preprint

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“…Remark 6.7. For a generalization of Theorem 6.2 to the case of arbitrary slope see our next paper [26].…”

Section: It Follows Thatmentioning

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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Twisted motivic Chern class and stable envelopes

Cited by 3 publications

References 21 publications

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

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

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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