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Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties

Author: Konstanze Rietsch
Journal: J. Amer. Math. Soc. 16 (2003), 363-392
MSC (2000): Primary 20G20, 15A48, 14N35, 14N15
Published electronically: November 29, 2002
Erratum: J. Amer. Math. Soc. 21 (2008), 611-614.
MathSciNet review: 1949164
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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

Keywords: Flag varieties, quantum cohomology, total positivity
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.
Article copyright: © Copyright 2002 American Mathematical Society