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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Polynomial bases: Positivity and Schur multiplication


Author: Dominic Searles
Journal: Trans. Amer. Math. Soc. 373 (2020), 819-847
MSC (2010): Primary 05E05; Secondary 05E10
DOI: https://doi.org/10.1090/tran/7670
Published electronically: October 28, 2019
MathSciNet review: 4068251
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Abstract: We establish a poset structure on combinatorial bases of multivariate polynomials defined by positive expansions and study properties common to bases in this poset. Included are the well-studied bases of Schubert polynomials, Demazure characters, and Demazure atoms; the quasi-key, fundamental, and monomial slide bases introduced in 2017 by Assaf and the author; and a new basis we introduce completing this poset structure. We show the product of a Schur polynomial and that an element of a basis in this poset expands positively in that basis; in particular, we give the first Littlewood-Richardson rule for the product of a Schur polynomial and a quasi-key polynomial. This rule simultaneously extends Haglund, Luoto, Mason, and van Willigenburg’s (2011) Littlewood-Richardson rule for quasi-Schur polynomials and refines their Littlewood-Richardson rule for Demazure characters. We also establish bijections connecting combinatorial models for these polynomials including semi-skyline fillings and quasi-key tableaux.


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Additional Information

Dominic Searles
Affiliation: Department of Mathematics and Statistics, University of Otago, Dunedin 9016, New Zealand
MR Author ID: 1039513
Email: dominic.searles@otago.ac.nz

Keywords: Schur polynomials, Demazure atoms, quasi-key polynomials, slide polynomials
Received by editor(s): August 21, 2017
Received by editor(s) in revised form: July 30, 2018
Published electronically: October 28, 2019
Article copyright: © Copyright 2019 American Mathematical Society