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1. QSym over Sym has a stable basis

Author

Aaron Lauve, Sarah K. Mason, Department of Mathematics [Austin], University of Texas at Austin [Austin], Department of Mathematics, Wake Forest University, Billey, Sara and Reiner, and Victor

Subjects
Pure mathematics, General Computer Science, Constructive proof, Dimension (graph theory), Free module, Stanley symmetric function, 0102 computer and information sciences, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], Space (mathematics), 01 natural sciences, Theoretical Computer Science, Combinatorics, Symmetric polynomial, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Mathematics::Metric Geometry, Discrete Mathematics and Combinatorics, symmetric functions, 0101 mathematics, quasisymmetric functions, Mathematics::Representation Theory, Ring of symmetric functions, Mathematics, Mathematics::Combinatorics, inverting compositions, Power sum symmetric polynomial, Mathematics::Complex Variables, 010102 general mathematics, free modules, compositions, Basis (universal algebra), Complete homogeneous symmetric polynomial, 16. Peace & justice, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], Symmetric function, [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], Computational Theory and Mathematics, 010201 computation theory & mathematics, Elementary symmetric polynomial
Abstract

We prove that the subset of quasisymmetric polynomials conjectured by Bergeron and Reutenauer to be a basis for the coinvariant space of quasisymmetric polynomials is indeed a basis. This provides the first constructive proof of the Garsia―Wallach result stating that quasisymmetric polynomials form a free module over symmetric polynomials and that the dimension of this module is $n!$., Nous prouvons que le sous-ensemble des polynômes quasisymétriques conjecturé par Bergeron et Reutenauer pour former une base pour l'espace coinvariant des polynômes quasisymétriques est en fait une base. Cela fournit la première preuve constructive du résultat de Garsia―Wallach indiquant que les polynômes quasisymétriques forment un module libre sur les polynômes symétriques et que la dimension de ce module est $n!$.

Published
2010

2. Refinements of the Littlewood-Richardson rule

Author

Kurt Luoto, S. van Willigenburg, James Haglund, Sarah K. Mason, University of Pennsylvania [Philadelphia], University of Washington [Seattle], Davidson College, University of British Columbia (UBC), Krattenthaler, Christian and Strehl, Volker and Kauers, and Manuel

Subjects
Pure mathematics, General Computer Science, Mathematics::Number Theory, Schur's lemma, nonsymmetric Macdonald polynomials, Schur algebra, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], Schur's theorem, Theoretical Computer Science, Schur functions, Combinatorics, Schur decomposition, Mathematics::Quantum Algebra, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, Littlewood-Richardson rule, quasisymmetric functions, Littlewood–Richardson rule, Mathematics::Representation Theory, Schur product theorem, Mathematics, Mathematics::Combinatorics, Mathematics::Complex Variables, Mathematics::History and Overview, Schur polynomial, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], 05E05 (Primary), 05E10, 33D52 (Secondary), [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], key polynomials, Schur complement, Combinatorics (math.CO)
Abstract

In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and Schutzenberger) could be used to define a new basis for the ring of quasisymmetric functions we call Quasisymmetric Schur functions (QS functions for short). In this paper we develop the combinatorics of these polynomials futher, by showing that the product of a Schur function and a Demazure atom has a positive expansion in terms of Demazure atoms. As a by-product, using the fact that both a QS function and a Demazure character have explicit expressions as a positive sum of atoms, we obtain the expansion of a product of a Schur function with a QS function (Demazure character) as a positive sum of QS functions (Demazure characters). Our formula for the coefficients in the expansion of a product of a Demazure character and a Schur function into Demazure characters is similar to known results and includes in particular the famous Littlewood-Richardson rule for the expansion of a product of Schur functions in terms of the Schur basis., 16 pages; addressed referee comments; version as accepted by Transactions of the AMS

Published
2009

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