As applications, we can generalize two other Schur positivity results to the flagged Schur case. Namely, we give a generalization of the Littlewood-Richardson Rule, showing how a product of flagged skew Schur polynomials expands positively in the basis of key polynomials - given two simple but necessary conditions on the partition shapes. We also deduce a log concavity inequality, similar to a theorem of Lam, Postnikov, and Pylyavskyy.
Our key tools are the recently defined shuffle tableaux of Nguyen and
Pylyavskyy. The proof of our main result comes down to showing that
flagged shuffle tableaux assemble themselves into Demazure
crystals. In order to prove this fact, we develop a new, semi-local
set of axioms for proving that a subset of a Type A crystal
forms a Demazure crystal.
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