Anisotropic Young Diagrams and Jack Symmetric Functions

S. V. Kerov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 1, pp. 51-64

Résumé

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 $b=\Lambda\setminus\lambda$.