[Paper Review] Shifted Jack polynomials, binomial formula, and applications

Andreĭ Okounkov, Grigori Olshanski|ArXiv.org|Aug 23, 1996

Advanced Combinatorial Mathematics9 references20 citations

TL;DR

This paper establishes a binomial formula for shifted Jack polynomials, providing an explicit combinatorial expression using reverse tableaux and $ heta$-weights. The key contribution is a generating function identity linking shifted Jack polynomials to Jack polynomials evaluated at $1+x_i$, with applications to representation theory and asymptotic character theory via $ heta$-dimension formulas.

ABSTRACT

In this note we prove an explicit binomial formula for Jack polynomials and discuss some applications of it.

Motivation & Objective

• To derive an explicit binomial formula for shifted Jack polynomials in terms of symmetric functions and combinatorial tableaux.
• To establish a connection between shifted Jack polynomials and Jack polynomials evaluated at $1+x_i$.
• To provide a new proof of the duality and vanishing properties of shifted Jack polynomials using combinatorial identities.
• To apply the binomial formula to derive $ heta$-dimension formulas for skew shapes and asymptotic character theory.

Proposed method

• Use of the Harish-Chandra isomorphism to relate differential operators to shifted symmetric polynomials.
• Definition of shifted Jack polynomials $P^{*}_{ u}(x; heta)$ as the unique elements in $ar{ heta}$-symmetric algebras vanishing at all $ u' \neq \nu$ with $|\nu'| \geq |\nu|$.
• Application of a combinatorial formula involving reverse tableaux $T$ on a partition $\mu$, with entries weakly decreasing across rows and strictly decreasing down columns.
• Use of $\psi_T(\theta)$, the $\theta$-weight of a tableau, to weight the monomials $\prod_{s\in\mu}(x_{T(s)} - a'(s) + \theta l'(s))$ in the expansion of $P^{*}_{\mu}(x;\theta)$.
• Derivation of the main binomial theorem: $\frac{P_{\lambda}(1+x_1,\dots,1+x_n;\theta)}{P_{\lambda}(1,\dots,1;\theta)} = \sum_{\mu} \frac{P^{*}_{\mu}(\lambda;\theta) Q_{\mu}(x;\theta)}{(n\theta)_{\mu}}$, where $Q_{\mu} = \frac{H'(\mu)}{H(\mu)} P_{\mu}$.
• Asymptotic analysis via scaling $\lambda \to \kappa\lambda$, $\mu \to \kappa\mu$ as $\kappa \to \infty$, leading to an integral representation for $P_{\mu}(\lambda)$ in terms of beta functions and a product of differences.

Experimental results

Research questions

• RQ1 How can a binomial-type expansion be constructed for shifted Jack polynomials in terms of standard Jack polynomials and their duals?
• RQ2 What is the combinatorial structure of shifted Jack polynomials, and how does it generalize the theory of shifted Schur functions?
• RQ3 How do the $\theta$-dimension formulas for skew shapes arise from the binomial identity and duality properties?
• RQ4 Can the binomial formula be used to derive integral representations for Jack polynomials via asymptotic limits?
• RQ5 What are the implications of the binomial formula for the asymptotic character theory of $U(\infty)$ and $S(\infty)$?

Key findings

• The binomial formula $\frac{P_{\lambda}(1+x_1,\dots,1+x_n;\theta)}{P_{\lambda}(1,\dots,1;\theta)} = \sum_{\mu} \frac{P^{*}_{\mu}(\lambda;\theta) Q_{\mu}(x;\theta)}{(n\theta)_{\mu}}$ holds for all partitions $\lambda$ and $\mu$, with $Q_{\mu}(x;\theta) = \frac{H'(\mu)}{H(\mu)} P_{\mu}(x;\theta)$.
• The $\theta$-dimension of a partition $\lambda$ is given by $\operatorname{\theta-dim}\lambda = \frac{| olimits|\lambda|!}{H(\lambda)}$, where $H(\lambda) = \prod_{s\in\lambda}(a(s) + \theta l(s) + 1)$.
• The ratio of $\theta$-dimensions for skew shapes satisfies $\frac{\operatorname{\theta-dim}\lambda/\mu}{\operatorname{\theta-dim}\lambda} = \frac{P^{*}_{\mu}(\lambda;\theta)}{| olimits|\lambda|(|\lambda|-1)\cdots(|\lambda|-|\mu|+1)}$, providing a combinatorial interpretation of skew $\theta$-dimensions.
• An integral representation for $P_{\mu}(\lambda)$ is derived as $P_{\mu}(\lambda) = \frac{1}{C(\mu,n)} \frac{1}{V(\lambda)^{2\theta-1}} \int \dots \int P_{\mu}(\nu) V(\nu) \Pi(\lambda,\nu;\theta) \, d\nu$, valid for real $\lambda$ and $\theta > 0$, with $C(\mu,n) = \prod_{i\leq n-1} B(\mu_i + (n-i)\theta, \theta)$.
• The asymptotic limit $\kappa \to \infty$ of $P^{*}_{\mu}(\kappa\lambda;\theta)/\kappa^{| olimits|\mu|}$ yields $P_{\mu}(\lambda;\theta)$, confirming consistency with the standard Jack polynomial limit.
• The method provides a new, combinatorial proof of the vanishing property $P^{*}_{\mu}(\lambda;\theta) = 0$ unless $\mu \subset \lambda$, and of the leading term identity $P^{*}_{\mu}(x;\theta) = P_{\mu}(x;\theta) + \text{lower degree terms}$.