On the Garsia Lie Idempotent
Canadian mathematical bulletin, Tome 48 (2005) no. 3, pp. 445-454
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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 ${{S}_{n}}$ .
Patras, Frédéric; Reutenauer, Christophe; Schocker, Manfred. On the Garsia Lie Idempotent. Canadian mathematical bulletin, Tome 48 (2005) no. 3, pp. 445-454. doi: 10.4153/CMB-2005-041-x
@article{10_4153_CMB_2005_041_x,
author = {Patras, Fr\'ed\'eric and Reutenauer, Christophe and Schocker, Manfred},
title = {On the {Garsia} {Lie} {Idempotent}},
journal = {Canadian mathematical bulletin},
pages = {445--454},
year = {2005},
volume = {48},
number = {3},
doi = {10.4153/CMB-2005-041-x},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2005-041-x/}
}
TY - JOUR AU - Patras, Frédéric AU - Reutenauer, Christophe AU - Schocker, Manfred TI - On the Garsia Lie Idempotent JO - Canadian mathematical bulletin PY - 2005 SP - 445 EP - 454 VL - 48 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2005-041-x/ DO - 10.4153/CMB-2005-041-x ID - 10_4153_CMB_2005_041_x ER -
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