Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914887433715712 |
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| author | Koncki, Jakub Weber, Andrzej |
| author_facet | Koncki, Jakub Weber, Andrzej |
| contents | Let $G$ be a linear semisimple algebraic group and $B$ its Borel subgroup. Let $\mathbb{T}\subset B$ 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 $K_\mathbb{T}(G/B)[y]$, 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 $K_\mathbb{T}(G/B)[y]$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_12746 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes Koncki, Jakub Weber, Andrzej Algebraic Geometry Combinatorics Primary 14C17, 14M15, Secondary 19L47, 20C08, 14N15 Let $G$ be a linear semisimple algebraic group and $B$ its Borel subgroup. Let $\mathbb{T}\subset B$ 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 $K_\mathbb{T}(G/B)[y]$, 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 $K_\mathbb{T}(G/B)[y]$ 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. |
| title | Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes |
| topic | Algebraic Geometry Combinatorics Primary 14C17, 14M15, Secondary 19L47, 20C08, 14N15 |
| url | https://arxiv.org/abs/2301.12746 |