Interpolation Polynomials, Binomial Coefficients, and Symmetric Function Inequalities

TL;DR

This work develops a comprehensive interpolation-polynomial framework for type A and BC, introducing four families ${A\mathrm{J}}, {A\mathrm{M}}, {B\mathrm{J}}, {B\mathrm{M}}$ whose top-degree terms recover Jack/Macdonald polynomials and whose binomial/LR coefficients generalize classical coefficients. The authors prove monotonicity and positivity for binomial coefficients, derive weighted-sum formulas for both binomial and Littlewood–Richardson coefficients, and establish integrality results under suitable parametrizations. A central accomplishment is a new symmetric-function duality: Jack positivity with respect to $\Omega_\lambda({\bf1}+x;\tau)$ is dual to the partition containment order, generalizing dualities known for Schur/Macdonald bases and connecting to existing CGS/KT inequalities. The results yield a Jack-polynomial–type positivity phenomenon in containment order and open avenues for Macdonald- and non-symmetric extensions, with potential applications in representation theory and combinatorial algebraic geometry.

Abstract

Interpolation polynomials were introduced by Knop--Sahi in type $A$, and Okounkov in type $BC$. They are inhomogeneous polynomials whose top terms are Jack and Macdonald polynomials. Thus the expansion coefficients for the product of two interpolation polynomials, known as Littlewood--Richardson coefficients, generalize the corresponding coefficients for Jack/Macdonald polynomials. Special values of interpolation polynomials, known as binomial coefficients, arise in the binomial type expansions of Jack/Macdonald polynomials and Koornwinder polynomials. We prove a number of results for interpolation polynomials and the associated coefficients. These include positivity and monotonicity results for binomial coefficients, partial positivity results for Littlewood--Richardson coefficients, and weighted sum formulas for both kinds of coefficients. As a special case of our results we obtain a new symmetric function inequality, which establishes a ``duality'' between Jack expansion positivity for symmetric functions, and the containment order on partitions, with respect to the shifted basis $Ω_λ({\bf1}+x;τ)$, where ${\bf1} =(1,\ldots,1)$ and $Ω_λ(x;τ)=P_λ(x;τ)/P_λ({\bf1};τ)$ is the normalized Jack polynomial. Our inequality can be seen as an analog of the inequalities of Cuttler--Greene--Skandera+Sra and Khare--Tao, which establish similar dualities between evaluation positivity on the positive orthant, and the dominance and weak dominance orders on partitions, with respect to the normalized Schur basis $Ω_λ(x)=s_λ(x)/s_λ({\bf1})$ and its shifted version $Ω_λ({\bf1}+x)$, respectively. In contrast to our result, the Jack versions of the two latter inequalities, although expected to hold, have not yet been proved.

Theorems & Definitions (80)

• Definition 1
• Theorem A
• Theorem B: Monotonicity
• Theorem C: Positivity
• Theorem D: Weighted Sum Formula
• Theorem E
• Theorem F: Adjacent Positivity for LR Coefficients
• Theorem G: Integrality and Positivity
• Remark 1
• Proposition 1