A plethysm formula for $p\sb µ(\underline x)\circ h\sb \lambda(\underline x)$

Abstract

A previous paper by the author \ref["A new plethysm formula for symmetric functions", J. Algebraic Combin., submitted] expresses the plethysm of the power sum symmetric function and the complete symmetric function, $p_µ(x)\circ h_a(x)$, as a sum of Schur functions with coefficients that are roots of unity. The paper under review extends this result to $p_µ(x)\circ h_\lambda(x)$, where the complete symmetric function is indexed by a partition rather than an integer. Specifically, the author proves that for $µ$ a partition of $b$ and $\lambda$ a partition of $a$ with length $t$, $p_µ(x)\circ h_\lambda(x)=\sum_T\omega^{\operatorname{maj}_{µ^t}(T)} s_{\operatorname{sh}(T)}(x)$, where the sum is over semistandard tableaux of weight $\lambda_1^b\lambda_2^b\cdots\lambda_t^b$ and $\omega^{\operatorname{maj}_{µ^t}}(T)$ is a root of unity. The proof is inductive and employs an intermediate result proved using the jeu de taquin.

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