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  1. Home > Articles & Issues >
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  3. QSym over Sym has a ...
Conference paper

QSym over Sym has a stable basis

Aaron Lauve (1), Sarah K Mason (2)
(1) Department of Mathematics [Austin]
(2) Department of Mathematics
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Imported on
January 31, 2017
Published on
December 31, 2009
Last modified on
March 31, 2025
Volume
DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
Proceedings
DOI
10.46298/dmtcs.2866
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https://about.hal.science/hal-authorisation-v1
National Science Foundation
0603351
PostDoctoral Research Fellowship
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QSym over Sym has a stable basis

Aaron Lauve (1), Sarah K Mason (2)
(1) Department of Mathematics [Austin]
(2) Department of Mathematics
Abstract
English
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 $n!$.
French
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 $n!$.
Keywords
  • [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
  • [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
  • quasisymmetric functions
  • symmetric functions
  • free modules
  • compositions
  • inverting compositions
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