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Geometric properties of
the Kazhdan–Lusztig Schubert basis
Cristian Lenart, Changjian Su, Kirill Zainoulline and Changlong Zhong |
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Vol. 17 (2023), No. 2, 435–464
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Abstract |
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We study classes determined by the Kazhdan–Lusztig basis of the Hecke algebra in the -theory and hyperbolic cohomology theory of flag varieties. We first show that, in -theory, the two different choices of Kazhdan–Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hodge modules. In equivariant hyperbolic cohomology, we show that if the Schubert variety is smooth, then the class it determines coincides with the class of the Kazhdan–Lusztig basis; this property was known as the smoothness conjecture. For Grassmannians, we prove that the classes of the Kazhdan–Lusztig basis coincide with the classes determined by Zelevinsky’s small resolutions. These properties of the so-called KL Schubert basis show that it is the closest existing analogue to the Schubert basis for hyperbolic cohomology; the latter is a very useful testbed for more general elliptic cohomologies. |
Keywords
Schubert calculus, flag variety, $K$-theory, hyperbolic
cohomology, Hecke algebra, Kazhdan–Lusztig Schubert basis
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Mathematical Subject Classification
Primary: 14M15, 55N20
Secondary: 05E99, 19L47, 20C08
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Milestones
Received: 22 July 2021
Revised: 9 February 2022
Accepted: 4 April 2022
Published: 24 March 2023
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Authors |
| Cristian Lenart | |
| Department of Mathematics and
Statistics State University of New York at Albany Albany, NY United States |
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| Changjian Su | |
| Department of Mathematics University of Toronto Toronto, ON Canada |
Yau Mathematical Sciences
Center Tsinghua University Beijing China |
| Kirill Zainoulline | |
| Department of Mathematics and
Statistics University of Ottawa Ottawa, ON Canada |
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| Changlong Zhong | |
| Department of Mathematics and
Statistics State University of New York at Albany Albany, NY United States |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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