Positivity determines the quantum cohomology of the odd symplectic Grassmannian of lines

Abstract

Let $\text{IG}:=\text{IG}(2,2n+1)$ denote the odd symplectic Grassmannian of lines which is a horospherical variety of Picard rank 1. The quantum cohomology ring ${\text{QH}}^{*}(\text{IG})$ has negative structure constants. For $n\ge 3$, we give a positivity condition that implies the quantum cohomology ring ${\text{QH}}^{*}(\text{IG})$ is the only quantum deformation of the cohomology ring ${\mathrm{H}}^{*}(\text{IG})$ up to the scaling of the quantum parameter. This is a modification of a conjecture by Fulton.

Keywords:

• Positivity
• quantum deformation
• quantum cohomology

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 14N35
• Secondary: 14N15
• 14M15

Acknowledgments

I would like to thank an anonymous referee for identifying a gap in the argument for the $\left|\lambda \right|>2n$ case. I would also like to thank Leonardo Mihalcea for a very useful conversation.