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 10778926. 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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