Quantum Cohomology of Grassmannians and Total Positivity

TL;DR

This paper explores the quantum cohomology of Grassmannians, connecting it to total positivity, providing explicit formulas, and proving inequalities for Schur polynomials at roots of unity.

Contribution

It offers a new proof of Peterson's isomorphism, constructs the totally positive part of a subvariety, and derives positivity-based inequalities for Schur polynomials.

Findings

Explicit formulas for Schubert basis elements on totally positive points

A new proof of the Vafa-Intriligator-Bertram formula for Gromov--Witten invariants

Positivity of Gromov--Witten invariants implies inequalities for Schur polynomials

Abstract

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.