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On the Garsia Lie Idempotent

Published online by Cambridge University Press:  20 November 2018

Frédéric Patras ,
Christophe Reutenauer  and
Manfred Schocker
Frédéric Patras
Affiliation:
Laboratoire J.-A. Dieudonné, Université de Nice, Parc Valrose, 06108 Nice, cedex 02, France
Christophe Reutenauer
Affiliation:
Université du Québec á Montréal, Département de mathématiques, CP 8888, succ. Centre-Ville, Montréal, QC, H3C 3P8 e-mail: christo@math.uqam.ca
Manfred Schocker
Affiliation:
Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, Wales, U.K.
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Abstract

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The orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group ${{S}_{n}}$, in each homogenous degree $n$. We give various characterizations of this Lie idempotent and show that it is uniquely determined by a certain unit in the group algebra of ${{S}_{n-1}}$. The inverse of this unit, or, equivalently, the Gram matrix of the orthogonal projection, is described explicitly. We also show that the Garsia Lie idempotent is not constant on descent classes (in fact, not even on coplactic classes) in ${{S}_{n}}$.

Keywords

17B0105A9916S3017B60
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 3 , 01 September 2005 , pp. 445 - 454
Copyright
Copyright © Canadian Mathematical Society 2005

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