Anisotropic Young diagrams and Jack symmetric functions
| dc.creator | Kerov, Sergei | |
| dc.date | 1997-12-27 | |
| dc.date.accessioned | 2026-07-07T05:23:26Z | |
| dc.date.available | 2026-07-07T05:23:26Z | |
| dc.description | We study the Young graph with edge multiplicities arising in a Pieri-type formula for Jack symmetric polynomials $P_μ(x;a)$ 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 $\sumν(c(b)+u)(c(b)+v)\dimν/\dimμ=(n+1)(n+uv)$, where $ν$ runs over immediate successors of a Young diagram $μ$ 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 $P_μ(x;a)$, it makes sense to consider `anisotropic' Young diagrams made of rectangular boxes of size $1\times a$. | |
| dc.description | 16 pages, AmSTeX, uses EPSF, three EPS figures | |
| dc.identifier | https://arxiv.org/abs/math/9712267 | |
| dc.identifier | http://arxiv.org/abs/math/9712267 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/76436 | |
| dc.subject | Combinatorics | |
| dc.subject | 05E05 05E10 33C45 | |
| dc.title | Anisotropic Young diagrams and Jack symmetric functions | |
| dc.type | text |