An Involution on the Quantum Cohomology Ring of the Grassmannian

TL;DR

This paper identifies an explicit involution on the quantum cohomology ring of the Grassmannian, combining cyclic action and Poincare duality, revealing new algebraic symmetries.

Contribution

It explicitly describes the involution on Schubert classes in the Grassmannian's quantum cohomology ring, connecting cyclic actions and Poincare duality.

Findings

The involution is an explicit transformation on Schubert classes.

It combines cyclic action with Poincare duality.

The involution aligns with previous observations by Postnikov.

Abstract

For a Fano manifold M, complex conjugation defines a real involution on the quantum cohomology ring. For the Grassmannian we identify this involution with an explicit transformation on Schubert classes defined over the integers. It is a composition of a power of the cyclic action recently defined by Agnihotri and Woodward with Poincare duality. The existence of the involution has been observed independently and from a different point of view by Postnikov (math.CO/0205165).