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arXiv · 1706.00744

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

Changzheng Li·Leonardo C. Mihalcea·Ryan Shifler
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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.

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Changzheng Li, Leonardo C. Mihalcea, Ryan Shifler. 2017-06-02. Conjecture $\mathcal{O}$ holds for the odd symplectic Grassmannian. https://doi.org/10.1112/blms.12268

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