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arxiv 2401.15640 v2 pith:M7Q2SN4E submitted 2024-01-28 math.AG math.CO

A Pl\"ucker coordinate mirror for partial flag varieties and quantum Schubert calculus

classification math.AG math.CO
keywords mirrorquantumbulletmathbbmathcalcalculuscohomologyconjecture
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We construct a Pl\"ucker coordinate superpotential $\mathcal{F}_-$ that is mirror to a partial flag variety $\mathbb{ F}\ell(n_\bullet)$. Its Jacobi ring recovers the small quantum cohomology of $\mathbb{ F}\ell(n_\bullet)$ and we prove a folklore conjecture in mirror symmetry. Namely, we show that the eigenvalues for the action of the first Chern class $c_1(\mathbb{ F}\ell(n_\bullet))$ on quantum cohomology are equal to the critical values of $\mathcal{F}_-$. We achieve this by proving new identities in quantum Schubert calculus that are inspired by our formula for $\mathcal{F}_-$ and the mirror symmetry conjecture.

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  1. Gamma conjecture I for flag varieties

    math.AG 2025-01 unverdicted novelty 7.0

    Proves Gamma conjecture I for flag varieties by showing the totally positive part of the Rietsch mirror matches the hat-Gamma class and contains the critical point for the Perron-Frobenius eigenvalue.