Relating the type A alcove path model to the right key of a semistandard Young tableau, with Demazure character consequences

TL;DR

This paper establishes a bijective correspondence between the alcove path model and the right key of semistandard Young tableaux, showing their equivalence in generating Demazure characters in type A.

Contribution

It proves that the final permutation from the alcove path model matches the right key of the tableau, unifying two combinatorial methods for Demazure character computation.

Findings

The alcove path model's final permutation corresponds bijectively to the tableau's right key.

The generating sets for Demazure characters from both methods are equivalent.

This correspondence simplifies understanding the combinatorial structures underlying Demazure characters.

Abstract

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 $T$ and outputs a saturated chain in the Bruhat order. The final permutation in this chain determines a family of Demazure characters for which $T$ contributes its weight. Separately, the right key of $T$ introduced by Lascoux and Sch\"utzenberger also determines a family of Demazure characters for which $T$ contributes its weight. In this paper we show that the final permutation in the chain produced by the alcove model corresponds bijectively to the right key of the tableau. From this it follows that the generating sets for the Demazure characters produced by these two methods are equivalent.