The authors prove the following conjecture of the reviewer [Adv. Math. 110, 234-246 (1995; Zbl 0823.05059)]: if the symmetric functions \(q_n\) are defined by: \(\prod_{n \geq 1} (1-q_nt^n)^{-1} = \sum_{n\geq 0} h_nt^n\), where the \(h_n\) are the complete symmetric functions, then \(-q_n\) is an \(\mathbb{N}\)-linear combination of Schur functions for \(n\geq 2\). If \(\lambda\) is a partition, denote by \(q_\lambda\) the corresponding product of \(q_n\), and \((r_\lambda)\) the basis adjoint to the basis \((q_\lambda)\) of symmetric functions.
Theorem: If \(\lambda = 1^{k_1} 2^{k_2} \dots\), then \(r_\lambda = (h_{k_1} \circ p_1)\) \((h_{k_2} \circ p_2) \dots\), where \(p_n\) is the \(n\)-th power sum and \(\circ\) denotes plethysm.
The authors interpret an integrality criterion of A. Dress [J. Algebra 101, 350-364 (1986; Zbl 0592.20012)] with the functions \(r_\lambda\). They also show that the following orthogonality relation holds: \[ \sum_{d|n} \ell_d^{(k)} \circ q_{n/d} = p_n \text{ if } n |k, \quad = 0 \text{ otherwise}, \] where \(\ell_n^{(k)}\) is the characteristic symmetric function of the representation of \(S_n\) induced by the representation of the subgroup generated by \(\gamma = (12 \dots n)\) defined by \(\gamma \mapsto \exp (2i \pi k/n)\) (the case \(k=1\) was obtained by the reviewer and \(\ell_n^{(1)}\) is the generating multivariate function of the \(n\)-th component of the free Lie algebra). The authors give a formula expressing \(r_\lambda\) in terms of Schur functions, involving ribbon tableaux.
Finally, they define noncommutative analogues of the \(q_\lambda\), and quasi-symmetric analogues of the \(r_\lambda\), dual to the latter.