A note on quantum products of Schubert classes in a Grassmannian

Anderson, Dave

Journal of Algebraic Combinatorics, Tome 25 (2007) no. 3, pp. 349-356.

Résumé

Summary: 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 $\sigma _{\lambda }\bigstar \sigma _{\mu }$. To do this, we show that for any two partitions $\lambda $and $\mu $, contained in a $k \times $( n - k) rectangle and such that the $180 ^{\deg }$-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.

Détail

Keywords: keywords quantum cohomology, toric tableau, Littlewood-Richardson number