Relating the right key to the type A filling map and minimal defining chains

There are several combinatorial methods that can be used to produce type A Demazure characters (key polynomials). The alcove path model of Lenart and Postnikov provides a procedure that inputs a semistandard tableau TT and outputs a saturated chain in the Bruhat order. The final permutation in this Bruhat chain determines a family of Demazure characters for which TT contributes its weight. Separately, the right key of TT introduced by Lascoux and Schützenberger also determines a family of Demazure characters for which TT contributes its weight. In this paper we show that the final permutation in this Bruhat chain corresponds bijectively to the right key of the tableau. We will also show that the minimal defining sequence for TT, as introduced by Lakshmibai, Musili, and Seshadri, is a sub-chain of this Bruhat chain. Most notably their final elements coincide. From this it follows that the generating sets for the Demazure characters produced by these three methods are equivalent.