A note on quantum products of Schubert classes in a Grassmannian

Abstract

Given two Schubert classes ${\sigma }_{\lambda }$ and ${\sigma }_{\mu }$ in the quantum cohomology of a Grassmannian, we construct a partition $\nu$, depending on $\lambda$ and $\mu$, such that ${\sigma }_{\nu }$ appears with coefficient 1 in the lowest (or highest) degree part of the quantum product ${\sigma }_{\lambda }\star {\sigma }_{\mu }$. To do this, we show that for any two partitions $\lambda$ and $\mu$, contained in a $k$-by-$(n-k)$ rectangle and such that the 180-degree rotation of one does not overlap the other, there is a third partition $\nu$, also contained in the rectangle, such that the Littlewood-Richardson number $c_{\lambda \mu }^{\nu }$ is 1.

9 pages, 4 figures