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Minimum Quantum Degrees for Isotropic Grassmannians in Types B and C

We give a formula in terms of Young diagrams to calculate the minimum positive integer $d$ such that $q^d$ appears in the quantum product of two Schubert classes for the submaximal isotropic Grassmannians in Types B and C. We do this by studying curve neighborhoods. We compute curve neighborhoods in several combinatorial models including $k$-strict partitions and a set of partitions where their inclusion is compatible with the Bruhat order.