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In this paper we pose the question of whether the (generalized) Mukaiinequalities hold for compact, positive monotone symplectic manifolds. We firstprovide a method that enables one to check whether the (generalized) Mukaiinequalities hold true. This only makes use of the almost complex structure ofthe manifold and the analysis of the zeros of the so-called generalized Hilbertpolynomial, which takes into account the Atiyah-Singer indices of all possibleline bundles. We apply this method to generalized flag varieties. In order tofind the zeros of the corresponding generalized Hilbert polynomial we introducea modified version of the Kostant game and study its combinatorial properties.

Generalizing the Mukai Conjecture to the symplectic category and the Kostant game