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JOURNALS // Funktsional'nyi Analiz i ego Prilozheniya // Archive

Funktsional. Anal. i Prilozhen., 2000 Volume 34, Issue 1, Pages 51–64 (Mi faa282)

This article is cited in 47 papers

Anisotropic Young Diagrams and Jack Symmetric Functions

S. V. Kerov

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: We study the Young lattice with the edge multiplicities $\varkappa_\alpha(\lambda,\Lambda)$ arising in the simplest Pieri formula for Jack symmetric polynomials $P_\lambda(x;\alpha)$ with parameter $\alpha$. A new proof of Stanley's $\alpha$-version of the hook formula is given. We also prove the formula
$$ \sum_\Lambda (c_\alpha(b)+u)(c_\alpha(b)+v)\varkappa_\alpha(\lambda,\Lambda)\varphi(\Lambda)= (n\alpha+uv)\varphi(\lambda), $$
where $\varphi(\lambda)=\prod_{b\in\lambda}(a(b)\alpha+l(b)+1)^{-1}$ and $c_\alpha(b)$ is the $\alpha$-contents of the new box $b=\Lambda\setminus\lambda$.

UDC: 519.217+517.986

Received: 05.05.1998

DOI: 10.4213/faa282


 English version:
Functional Analysis and Its Applications, 2000, 34:1, 41–51

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