[Paper Review] Difference equations and symmetric polynomials defined by their zeros

Friedrich Knop, Siddhartha Sahi|arXiv (Cornell University)|Oct 15, 1996

Advanced Topics in Algebra19 citations

TL;DR

This paper introduces a family of non-homogeneous symmetric polynomials defined by vanishing conditions at shifted partition points, parameterized by $ r \in \mathbb{C} $. It proves these polynomials are simultaneous eigenfunctions of $ n $ commuting difference operators, which generalize the Sekiguchi-Debiard differential operators, and shows their top homogeneous components are Jack polynomials. The key contribution is a new proof of the Pieri rule for Jack polynomials using these inhomogeneous polynomials and a conjecture on integrality and positivity of their integral forms.

ABSTRACT

In this paper, we introduce a new family of symmetric polynomials which depends on a parameter r. They are defined by specifying certain of their zeros. For the parameter values 1/2, 1, and 2 they have an interpretation in terms of Capelli identities. First, we give explicit formulas in some special cases. Then we show that the polynomials can also be defined in terms of difference equations. As a corollary we obtain that their top homogeneous part is a Jack polynomial. This is used to give a new proof of the Pieri formula for Jack polynomials.

Motivation & Objective

• To systematically analyze a class of non-homogeneous symmetric polynomials defined by vanishing conditions at shifted partitions.
• To establish that these polynomials are eigenfunctions of commuting difference operators, which are discrete analogues of the Sekiguchi-Debiard differential operators.
• To show that the top homogeneous component of each such polynomial is a Jack polynomial, linking the inhomogeneous family to classical symmetric function theory.
• To provide a new proof of the Pieri rule for Jack polynomials using the structure of these inhomogeneous polynomials.
• To formulate and support a conjecture on integrality and positivity of the integral form of the inhomogeneous polynomials, generalizing Macdonald's conjecture.

Proposed method

• Define symmetric polynomials $ P_{ ho}^{r\delta} $ via a vanishing condition: $ P_{ ho}^{r\delta}(\mu + r\delta) = 0 $ for all $ \mu \neq \rho $ with $ |\mu| \leq |\rho| $.
• Use a recursive interpolation construction based on induction on $ n $ and $ d $, expressing the polynomial as a sum of a shifted symmetric function in $ n-1 $ variables and a product term involving $ \prod (x_i - \varrho_n) $.
• Introduce difference operators $ \mathcal{E}_k $ that act as discrete analogues of the Sekiguchi-Debiard differential operators, and prove they commute and act diagonally on the $ P_{\lambda}^{r\delta} $.
• Establish the eigenvalue relation $ \mathcal{E}_k P_{\mu}^{r\delta} = \sum_{\lambda} \psi'_{\lambda/\mu}(1/r) P_{\lambda}^{r\delta} $, where the sum is over partitions $ \lambda $ differing from $ \mu $ by a vertical $ k $-strip.
• Derive explicit formulas for the eigenvalues $ \psi'_{\lambda/\mu}(1/r) $ using combinatorial identities involving the $ c_{\lambda}^{r\delta} $ coefficients and cancellation patterns in the product expressions.
• Define the integral form $ J_{\lambda}^{r\delta}(x) = (-1)^{|\lambda|} c_{\lambda}(1/r) P_{\lambda}^{r\delta}(-x) $, and conjecture that its expansion in monomial symmetric functions has positive integral coefficients in $ \alpha = 1/r $.

Experimental results

Research questions

• RQ1 How can symmetric polynomials be systematically constructed via vanishing conditions at shifted partitions, especially when non-homogeneous?
• RQ2 What is the structure of the difference operators that act diagonally on these inhomogeneous symmetric polynomials?
• RQ3 How do the eigenvalues of these difference operators relate to known combinatorial rules, such as the Pieri rule?
• RQ4 What integrality and positivity properties do the integral forms of these polynomials exhibit?
• RQ5 Can the inhomogeneous polynomials provide a new proof of classical results like the Pieri rule for Jack polynomials?

Key findings

• The polynomials $ P_{\lambda}^{r\delta} $ are uniquely defined up to scalar by the vanishing condition $ P_{\lambda}^{r\delta}(\mu + r\delta) = 0 $ for all $ \mu \neq \lambda $ with $ |\mu| \leq |\lambda| $, under the $ d $-dominance condition on $ r\delta $.
• The $ n $ difference operators $ \mathcal{E}_k $ commute and act diagonally on the $ P_{\lambda}^{r\delta} $, with eigenvalues given by $ \psi'_{\lambda/\mu}(1/r) $, which are rational functions in $ r $.
• The top homogeneous component of $ P_{\lambda}^{r\delta} $ is a Jack polynomial, establishing a direct link between the inhomogeneous and classical homogeneous families.
• A new proof of the Pieri rule for Jack polynomials is obtained by analyzing the action of $ \mathcal{E}_k $ on $ P_{\mu}^{r\delta} $, showing the structure of vertical $ k $-strip insertions.
• The integral form $ J_{\lambda}^{r\delta} $ is conjectured to have expansion coefficients in $ \alpha = 1/r $ that are positive integers, generalizing Macdonald’s conjecture for the homogeneous case.
• The authors have recently proven the integrality part of this conjecture, and the positivity part is supported by extensive computations.