Research Article

Galkin’s lower bound conjecture holds for the Grassmannian

Published: 2020-2-1

Journal: Communications in Algebra

DOI: 10.1080/00927872.2019.1663383

Abstract

Abstract Let Gr(k,n) be the Grassmannian. The quantum multiplication by the first Chern class c1(Gr(k,n)) induces an endomorphism ˆc1 of the finite-dimensional vector space QH*(Gr(k,n))|q=1 specialized at q = -1. Our main result is a case that a conjecture by Galkin holds. It states that the largest real eigenvalue of ĉ1 is greater than or equal to dim Gr(k,n) + 1 with equality if and only if Gr(k,n) = Pn−1. Keywords: Algebraic geometry, conjecture O, quantum cohomology 2010 Mathematics Subject Classification: Primary 14N35, Secondary 15B48, 14N15, 14M15 Acknowledgments The third named author thanks Leonardo Mihalcea for useful discussions. We thank the anonymous referees for their comments and an alternative proof.

Faculty Members

  • Ryan M. Shifler - Department of Mathematics and Computer Science, Salisbury University, Salisbury, MD, USA
  • Laura Short - Department of Mathematics and Computer Science, Salisbury University, Salisbury, MD, USA
  • La’Tier Evans - Department of Mathematics and Computer Science, Salisbury University, Salisbury, MD, USA
  • Lisa Schneider - Department of Mathematics and Computer Science, Salisbury University, Salisbury, MD, USA
  • Stephanie Warman - Department of Mathematics and Computer Science, Salisbury University, Salisbury, MD, USA

Themes

  • Chern Class
  • Algebraic Geometry
  • Grassmannian
  • Conjectures in Mathematics
  • Eigenvalue Analysis
  • Quantum Cohomology

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