Bergeron, F.; Bergeron, N.; Garsia, A. M. 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) Cited in 1 ReviewCited in 21 Documents MSC: 17B01 Identities, free Lie (super)algebras 20C30 Representations of finite symmetric groups 05E10 Combinatorial aspects of representation theory Keywords:representations of symmetric group; free Lie algebra; idempotents; P- partitions; dimension formula Citations:Zbl 0694.00010 PDFBibTeX XML