On the Garsia Lie Idempotent

http://dx.doi.org/10.4153/CMB-2005-041-x
Canad. Math. Bull. 48(2005), 445-454

  Published:2005-09-01
 Printed: Sep 2005

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 $S_n$.