Abstract
We prove that the sheaf Euler characteristic of the product of a Schubert class and an opposite Schubert class in the quantum K-theory ring of a (generalized) flag variety G/P is equal to \(q^d\), where d is the smallest degree of a rational curve joining the two Schubert varieties. This implies that the sum of the structure constants of any product of Schubert classes is equal to 1. Along the way, we provide a description of the smallest degree d in terms of its projections to flag varieties defined by maximal parabolic subgroups.
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The authors acknowledge support from NSF Grant DMS-1503662 (Buch), NSFC Grants 11771455, 11822113, 11831017, and Guangdong Introducing Innovative and Enterpreneurial Teams No. 2017ZT07X355 (Li), and the NSA Young Investigator Award H98320-16-1-0013 and a Simons Collaboration Grant (Mihalcea).
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Buch, A.S., Chung, S., Li, C. et al. Euler characteristics in the quantum K-theory of flag varieties. Sel. Math. New Ser. 26, 29 (2020). https://doi.org/10.1007/s00029-020-00557-7
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DOI: https://doi.org/10.1007/s00029-020-00557-7