Monotonicity for generalized binomial coefficients and Jack positivity

2025/08/07 by Hong Chen, Siddhartha Sahi, Chen, Hong +1
Mathematics · #Advanced Mathematical Identities #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2508.05759

openalex publication_date 2025/08/07 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

Binomial formulas for Schur polynomials and Jack polynomials were studied by Lascoux in 1978, and Kaneko, Okounkov--Olshanski and Lassalle in the 1990s. We prove that the associated binomial coefficients are monotone and derive some symmetric function inequalities, in particular, a Schur positivity and Jack positivity result. These inequalities are similar to those studied by Newton, Muirhead, Gantmacher, Cuttler--Greene--Skandera, Sra and Khare--Tao.

Citations

• Interpolation Polynomials, Binomial Coefficients, and Symmetric Function Inequalities
• Positive formula for Jack polynomials, Jack characters and proof of Lassalle's conjecture
• The uses of random partitions
• The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture
• The binomial formula for nonsymmetric Macdonald polynomials
• (Shifted) Macdonald Polynomials: q-Integral Representation and Combinatorial Formula

Related

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• Interpolation Polynomials, Binomial Coefficients, and Symmetric Function Inequalities
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• Schur polynomials and matrix positivity preservers