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Idempotents for the free Lie algebra and q-enumeration. (English) Zbl 0721.17006

Invariant theory and tableaux, Proc. Workshop, Minneapolis/MN (USA) 1988, IMA Vol. Math. Appl. 19, 166-190 (1990).
[For the entire collection see Zbl 0694.00010.]
Let F(A) denote the free Lie algebra over an alphabet A. The n-th homogeneous component of F(A) may be expressed as the linear span of polynomials obtained by multiplying words of length n by a fixed idempotent of the group algebra of \(S_ n\). The authors construct such idempotents and prove some of their properties making use of the theory of P-partitions. In particular they construct an idempotent whose coefficients give the Taylor expansion of the reciprocal of the cyclotomic polynomial. They also give a new proof of the dimension formula for F(A) and prove a conjecture of R. Stanley concerning certain representations of \(S_ n\).
Reviewer: M.Boral (Adana)

MSC:

17B01 Identities, free Lie (super)algebras
20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory

Citations:

Zbl 0694.00010