Key polynomials and a flagged Littlewood—Richardson rule

AbstractThis paper studies a family of polynomials called key polynomials, introduced by Demazure and investigated combinatorially by Lascoux and Schützenberger. We give two new combinatorial interpretations for these key polynomials and show how they provide the connection between two relatively recent combinatorial expressions for Schubert polynomials. We also give a flagged Littlewood—Richardson rule, an expansion of a flagged skew Schur function as a nonnegative sum of key polynomials

A Direct Way to Find the Right Key of a Semistandard Young Tableau

The right and left key of a semistandard Young tableau were introduced by Lascoux and Schutzenberger in 1990. Most prominently, the right key is a tool used to find Demazure characters for sl(n,C). Previous methods used to compute these keys require introducing other types of combinatorial objects. This paper gives methods to obtain the right and left keys by inspection of the semistandard Young tableau.Comment: 8 pages, 1 figure. Virtually identical to the version submitted in July 2011. To be contained in the author's doctoral thesis written under the supervision of Robert A. Procto

Flagged Skew Schur Polynomials Twisted By Roots Of Unity

We generalize a theorem of Littlewood concerning the factorization of Schur polynomials when their variables are twisted by roots of unity. We show that a certain family of flagged skew Schur polynomials admit a similar factorization. These include an interesting family of Demazure characters as a special case

A Direct Way to Find the Right Key of a Semistandard Young Tableau

Abstract The right key of a semistandard Young tableau is a tool used to find Demazure characters for {sl_n(\mathbb{C})}$ . This paper gives methods to obtain the right and left keys by inspection of the semistandard Young tableau

Schubert polynomials as integer point transforms of generalized permutahedra

We show that the dual character of the flagged Weyl module of any diagram is a positively weighted integer point transform of a generalized permutahedron. In particular, Schubert and key polynomials are positively weighted integer point transforms of generalized permutahedra. This implies several recent conjectures of Monical, Tokcan and Yong.Comment: 8 pages. Corrected title in arXiv metadata (d'oh); no change to manuscrip