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August 2021 Maximal Newton Points and the Quantum Bruhat Graph
Elizabeth Milićević
Michigan Math. J. 70(3): 451-502 (August 2021). DOI: 10.1307/mmj/20175356


We discuss a surprising relationship between the partially ordered set of Newton points associated with an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the unique maximum element in this poset in terms of paths in the quantum Bruhat graph, whose vertices are indexed by elements in the finite Weyl group. Key to establishing this connection is the fact that paths in the quantum Bruhat graph encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson’s isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. One important geometric application of the present work is an inequality which provides a necessary condition for nonemptiness of certain affine Deligne–Lusztig varieties in the affine flag variety.


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Elizabeth Milićević. "Maximal Newton Points and the Quantum Bruhat Graph." Michigan Math. J. 70 (3) 451 - 502, August 2021.


Received: 22 March 2017; Revised: 10 August 2020; Published: August 2021
First available in Project Euclid: 23 December 2020

Digital Object Identifier: 10.1307/mmj/20175356

Primary: 11G25 , 20G25
Secondary: 06A11 , 14N15 , 20F55

Rights: Copyright © 2021 The University of Michigan


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Vol.70 • No. 3 • August 2021
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