Representations of the symmetric group in the free Lie (super-)algebra and in the space of harmonic polynomials (Q1086350)

The authors compute the characters of the natural representations of the symmetric group \(S_ n\) in the free Lie algebra \({\mathfrak g}_ n\) with n generators and in the free Lie superalgebra \({\mathfrak g}_ n'\) with n odd generators. It is obtained by describing the structure of representations of \(S_ n\) in subspaces of harmonic polynomials \({\mathcal H}_ n\) of the form \(\oplus_{k\equiv m(mod r)}{\mathcal H}^ k_ n\), where r divides n, m is a residue modulo r.