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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties
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by Konstanze Rietsch PDF
J. Amer. Math. Soc. 16 (2003), 363-392 Request permission

Abstract:

We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in $GL_n$ forms a real semi-algebraic cell of dimension $n-1$. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of $GL_n(\mathbb {C})$ relying in particular on the positivity of the structure constants, which are enumerative Gromov–Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterson’s which we explain with proofs in the type $A$ case.
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Additional Information
  • Konstanze Rietsch
  • Affiliation: Mathematical Institute, University of Oxford, Oxford, United Kingdom
  • Address at time of publication: King’s College, University of London, London, United Kingdom
  • Email: rietsch@maths.ox.ac.uk, rietsch@mth.kcl.ac.uk
  • Received by editor(s): December 10, 2001
  • Received by editor(s) in revised form: September 14, 2002
  • Published electronically: November 29, 2002
  • Additional Notes: During a large part of this work the author was an EPSRC postdoctoral fellow (GR/M09506/01) and Fellow of Newnham College in Cambridge. The article was completed while supported by the Violette and Samuel Glasstone Foundation at Oxford.
  • © Copyright 2002 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 16 (2003), 363-392
  • MSC (2000): Primary 20G20, 15A48, 14N35, 14N15
  • DOI: https://doi.org/10.1090/S0894-0347-02-00412-5
  • MathSciNet review: 1949164