PREPRINT 24/1997

Sergei KEROV

ANISOTROPIC YOUNG DIAGRAMS AND JACK SYMMETRIC FUNCTIONS

This preprint was accepted December, 1997.
Contact: S.V.Kerov

ABSTRACT:
We study the Young graph with edge multiplicities
$\varkappa_\alpha(\lambda,\Lambda)$ arising in a Pieri-type
formula $p_1(x)\,P_\lambda(x;\alpha)= \sum_{\Lambda:\lambda\nearrow\Lambda} \varkappa_\alpha(\lambda,\Lambda)\,P_\Lambda(x;\alpha)$ for Jack
symmetric polynomials $P_\lambda(x;\alpha)$ with parameter
$\alpha$. Starting with $\dim_\alpha\varnothing=1$, we define
recurrently the numbers $\dim_\alpha\Lambda= \sum\varkappa_\alpha(\lambda,\Lambda)\,\dim_\alpha\lambda$, and
we set $\varphi(\lambda)= \prod_{b\in\lambda}\big(a(b)\alpha+l(b)+1\big)^{-1}$ (where
$a(b)$ and $l(b)$ are the arm- and leg-length of a box $b$).

New proofs are given for two known results. The first is the
$\alpha$-hook formula $\dim_\alpha\lambda=n!\,\alpha^n\, \prod_{b\in\lambda}\big((a(b)+1)\alpha+l(b)\big)^{-1}$. Secondly,
we prove (for all $u,v\in\Bbb{C}$) the summation formula
$\sum_{\Lambda:\lambda\nearrow\Lambda} (c_\alpha(b)+u)(c_\alpha(b)+v) \varkappa_\alpha(\lambda,\Lambda)\,\varphi(\Lambda)= (n\alpha+uv)\;\varphi(\lambda)$, where $c_\alpha(b)$ is the
$\alpha$-content of a new box $b=\Lambda\setminus\lambda$. This
identity implies the existence of an interesting family of
positive definite central functions on the infinite symmetric
group.

The approach is based on the interpretation of a Young diagram as
a pair of interlacing sequences, so that analytic techniques may
be used to solve combinatorial problems. We show that when dealing
with Jack polynomials $P_\lambda(x;\alpha)$, it makes sense to
consider {\it anisotropic Young diagrams} made of rectangular
boxes of size $1\times\alpha$.

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