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. doi:10.37236/1299. https://resolver.caltech.edu/CaltechAUTHORS:DORejc97
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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 | ||||||
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Additional Information: | Submitted: September 10, 1996; Accepted: May 2, 1997 | ||||||
Subject Keywords: | Symmetric Functions, Plethysm | ||||||
Issue or Number: | 1 | ||||||
DOI: | 10.37236/1299 | ||||||
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 | ||||||
Deposited By: | Archive Administrator | ||||||
Deposited On: | 26 Sep 2006 | ||||||
Last Modified: | 08 Nov 2021 20:22 |
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