# Conjecture $\mathcal{O}$ holds for the odd symplectic Grassmannian

## TL;DR

This paper proves that property $ ext{O}$, related to eigenvalues of quantum multiplication operators, holds for odd-symplectic Grassmannians, expanding understanding of quantum cohomology in this class of Fano manifolds.

## Contribution

The paper establishes property $ ext{O}$ for odd-symplectic Grassmannians using quantum Chevalley formulas and Perron-Frobenius theory, a novel application in this context.

## Key findings

- Property $ ext{O}$ holds for $ ext{IG}(k, 2n+1)$.
- Quantum Chevalley formula is key to the proof.
- Eigenvalue properties are confirmed for these manifolds.

## Abstract

Let $\mathrm{IG}(k, 2n+1)$ be the odd-symplectic Grassmannian. Property $\mathcal{O}$, introduced by Galkin, Golyshev and Iritani for arbitrary complex, Fano manifolds $X$, is a statement about the eigenvalues of the linear operator obtained by the quantum multiplication by the anticanonical class of $X$. We prove that property $\mathcal{O}$ holds in the case when $X= \mathrm{IG}(k, 2n+1)$ is an odd-symplectic Grassmannian. The proof uses the combinatorics of the recently found quantum Chevalley formula for $\mathrm{IG}(k, 2n+1)$, together with the Perron-Frobenius theory of nonnegative matrices.