Anisotropic Young diagrams and Jack symmetric functions

Abstract

We study the Young graph with edge multiplicities arising in a Pieri-type formula for Jack symmetric polynomials $P_\mu(x;a)$ with a parameter $a$. Starting with the empty diagram, we define recurrently the `dimensions' $\dim_a$ in the same way as for the Young lattice or Pascal triangle. New proofs are given for two known results. The first is the $a$-hook formula for $\dim_a$, first found by R.Stanley. Secondly, we prove (for all complex $u$ and $v$) a generalization of the identity $\sum\nu(c(b)+u)(c(b)+v)\dim\nu/\dim\mu=(n+1)(n+uv)$, where $\nu$ runs over immediate successors of a Young diagram $\mu$ with $n$ boxes. Here $c(b)$ is the content of a new box $b$. The identity is known to imply 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 $P_\mu(x;a)$, it makes sense to consider `anisotropic' Young diagrams made of rectangular boxes of size $1\times a$.