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Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

Published online by Cambridge University Press:  20 November 2018

Nantel Bergeron ,
Christophe Reutenauer ,
Mercedes Rosas  and
Mike Zabrocki
Nantel Bergeron
Affiliation:
Mathematics and Statistics, York University, 4700 Keele St., Toronto, ON, M5A 4T5 e-mail: bergeron@mathstat.yorku.ca, e-mail: rosas@mathstat.yorku.ca, e-mail: zabrocki@mathstat.yorku.ca
Christophe Reutenauer
Affiliation:
LaCIM, Université du Québec à Montréal, C.P. 8888, succursale Centre-ville, Montréal, QC, H3C 3P8 e-mail: christo@math.uqam.ca
Mercedes Rosas
Affiliation:
Mathematics and Statistics, York University, 4700 Keele St., Toronto, ON, M5A 4T5 e-mail: bergeron@mathstat.yorku.ca, e-mail: rosas@mathstat.yorku.ca, e-mail: zabrocki@mathstat.yorku.ca
Mike Zabrocki
Affiliation:
Mathematics and Statistics, York University, 4700 Keele St., Toronto, ON, M5A 4T5 e-mail: bergeron@mathstat.yorku.ca, e-mail: rosas@mathstat.yorku.ca, e-mail: zabrocki@mathstat.yorku.ca
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Abstract

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We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions. The bases for this algebra are indexed by set partitions. We show that there exists a natural inclusion of the Hopf algebra of noncommutative symmetric functions in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommutative polynomials and conclude two analogues of Chevalley’s theorem in the noncommutative setting.

Keywords

16W3005A1805E10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 2 , 01 April 2008 , pp. 266 - 296
Copyright
Copyright © Canadian Mathematical Society 2008

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