Quantum cohomology rings of Grassmannians and total positivity

Abstract

We give a proof of a result of D. Peterson identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of ${\rm GL}_n$. 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 C. Vafa, K. Intriligator, and A. 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.