On the Garsia Lie Idempotent

[摘要] The orthogonal projection of the free associative algebra onto thefree Lie algebra is afforded by an idempotent in the rational groupalgebra of the symmetric group $S_n$, in each homogenous degree$n$. We give various characterizations of this Lie idempotent and showthat it is uniquely determined by a certain unit in the group algebraof $S_{n-1}$. The inverse of this unit, or, equivalently, the Grammatrix of the orthogonal projection, is described explicitly. We alsoshow that the Garsia Lie idempotent is not constant on descent classes(in fact, not even on coplactic classes) in $S_n$.

[学科分类] 数学（综合）

[关键词] Toric variety;torus action;cohomology ring;simple polytope;polytope algebra