CaltechAUTHORS
A Caltech Library Service

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

Doran, William F., IV (1997) A plethysm formula for $p\sb µ(\underline x)\circ h\sb \lambda(\underline x)$. Electronic Journal of Combinatorics, 4 (1). R14. ISSN 1077-8926. https://resolver.caltech.edu/CaltechAUTHORS:DORejc97

 Preview
PDF - Published Version
See Usage Policy.

211Kb

Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:DORejc97

## 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.

Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.37236/1299DOIArticle
Additional Information:Submitted: September 10, 1996; Accepted: May 2, 1997
Subject Keywords:Symmetric Functions, Plethysm
Issue or Number:1
Record Number:CaltechAUTHORS:DORejc97
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:DORejc97
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:5067
Collection:CaltechAUTHORS