Mathematics – Combinatorics
Scientific paper
1997-12-27
Mathematics
Combinatorics
16 pages, AmSTeX, uses EPSF, three EPS figures
Scientific paper
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 $1\times a$.
No associations
LandOfFree
Anisotropic Young diagrams and Jack symmetric functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Anisotropic Young diagrams and Jack symmetric functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Anisotropic Young diagrams and Jack symmetric functions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-145438