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# On the Garsia Lie Idempotent

Published:2005-09-01
Printed: Sep 2005
• Frédéric Patras
• Christophe Reutenauer
• Manfred Schocker
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## Abstract

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 $S_n$.
 MSC Classifications: 17B01 - Identities, free Lie (super)algebras 05A99 - None of the above, but in this section 16S30 - Universal enveloping algebras of Lie algebras [See mainly 17B35] 17B60 - Lie (super)algebras associated with other structures (associative, Jordan, etc.) [See also 16W10, 17C40, 17C50]