Interpolation Polynomials, Binomial Coefficients, and Symmetric Function Inequalities

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 $\Omega_\lambda({\bf1}+x;\tau)$, where ${\bf1} =(1,\ldots,1)$ and $\Omega_\lambda(x;\tau)=P_\lambda(x;\tau)/P_\lambda({\bf1};\tau)$ 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 $\Omega_\lambda(x)=s_\lambda(x)/s_\lambda({\bf1})$ and its shifted version $\Omega_\lambda({\bf1}+x)$, respectively. In contrast to our result, the Jack versions of the two latter inequalities, although expected to hold, have not yet been proved.