Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Ryan M. Shifler

Abstract

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

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

Abstract

Let denote the odd symplectic Grassmannian of lines which is a horospherical variety of Picard rank 1. The quantum cohomology ring has negative structure constants. For , we give a positivity condition that implies the quantum cohomology ring is the only quantum deformation of the cohomology ring up to the scaling of the quantum parameter. This is a modification of a conjecture by Fulton.
Paper Structure (4 sections, 10 theorems, 17 equations)

This paper contains 4 sections, 10 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $|\lambda|=2n$ case
  4. The $|\lambda|>2n$ case

Key Result

Theorem 1.4

Let $n\geq 3$. Suppose that $\{\sigma_\lambda: \lambda \in \Lambda \}$ is a quantum deformation of the Schubert basis $\{\tau_\lambda: \lambda \in \Lambda \}$ of $\mathrm{QH}^*(\mathrm{IG})$ such that Condition (**) holds. Then $\tau_\lambda=\sigma_\lambda$ for all $\lambda \in \Lambda$.

Theorems & Definitions (23)

  • Definition 1.1
  • Remark 1.2
  • Conjecture 1
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • ...and 13 more