2024/03/04 by Hong Chen, Siddhartha Sahi, Chen, Hong +1 · 3 citations
Mathematics · #05E05 #Combinatorics (math.CO) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2403.02490
openalex publication_date 2024/03/04 · openalex created_date 2024/03/07 · openalex updated_date 2026/07/31
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 Ωλ(\bf1+x;τ), where \bf1 =(1,…,1) and Ωλ(x;τ)=Pλ(x;τ)/Pλ(\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 Ωλ(x)=sλ(x)/sλ(\bf1) and its shifted version Ωλ(\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.