Quantum Cohomology of Grassmannians and Total Positivity

Details Quantum Cohomology of Grassmannians and Total Positivity Quantum Cohomology of Grassmannians and Total Positivity

2001-12-03

arxiv.org/abs/math/0112022v1

Mathematics

Quantum Algebra

Revised version of ESI preprint 923. To appear in Duke Math J

Scientific paper

We give a proof of a result of D. Peterson's identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of $GL_n$. The totally positive part of this subvariety is then constructed and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of Vafa and Intriligator and Bertram for the structure constants (Gromov--Witten invariants). Finally, we use the positivity of these Gromov--Witten invariants to prove certain inequalities for Schur polynomials at roots of unity.

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