A new plethysm formula for symmetric functions
The paper derives a formula for the plethysm of a power sum symmetric function, \(p_\mu\), with a complete symmetric function with one part, \(h_a\) (also known as a Schur function with one part, \(s_a)\). The main result is, \[ p_\mu(x)\circ h_a(x)= \sum_T\omega^{\text{maj}_\mu(T)} s_{\text{sh} (T)}(x) \] where \(\mu\) is a partition of \(b\), the sum is over tableaux \(T\) with weight \(a^b,\omega\) is a root of unity, \(\text{sh}(T)\) is the shape of \(T\), and \(\text{maj}_\mu(T)\) is a major index-like statistic on tableaux. The proof is mainly algebraic and employs a key result from \textit{A. Lascoux}, \textit{B. Leclerc}, and \textit{J.-Y. Thibon} [Green polynomials and Hall-Littlewood functions at roots of unity, Eur. J. Comb. 15, No. 2, 173-180 (1994; Zbl 0789.05093)]. The main result has also been generalized by the author to the case of the plethysm of a power sum symmetric function, \(p_\mu\), with a complete symmetric function with an arbitrary number of parts, \(h_\lambda\), see \textit{W. F. Doran} IV [Electron. J. Comb. 4, No. 1, Research paper R14, 10 p. (1997; Zbl 0885.05108); printed version J. Comb. 4, No. 1, 137-146 (1997)].
scientific article; zbMATH DE number 1228561