The monomial expansions for modified Macdonald polynomials

2025/06/29 by Emma Yu Jin, Xiaowei Lin, Jin, Emma Yu +1
Mathematics · #05A19 #05E05 #33D52 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2506.23373

openalex publication_date 2025/06/29 · openalex created_date 2025/10/20 · openalex updated_date 2026/07/28

Abstract

We discover a family A of sixteen statistics on fillings of any given Young diagram and prove new combinatorial formulas for the modified Macdonald polynomials, that is, H_λ(X;q,t)=∑_{σ∈ T(λ)}x^σq^maj(σ)t^η(σ) for each statistic η∈ A. Building upon this new formula, we establish four compact formulas for the modified Macdonald polynomials, namely, H_λ(X;q,t)=∑_σd_ε(σ)x^σq^maj(σ)t^η(σ) which is summed over all canonical or dual canonical fillings of a Young diagram and d_ε(σ) is a product of t-multinomials. Finally, the compact formulas enable us to derive four explicit expressions for the monomial expansion of modified Macdonald polynomials, one of which coincides with the formula given by Garbali and Wheeler (2019).

Citations

• New formulas for Macdonald polynomials via the multispecies exclusion and zero range processes
• Probabilistic operators for non-attacking tableaux and a compact formula for the symmetric Macdonald polynomials
• Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials
• Modified Macdonald polynomials and integrability
• A combinatorial formula for Macdonald polynomials
• Hilbert schemes, polygraphs, and the Macdonald positivity conjecture
• New combinatorial formula for modified Hall-Littlewood polynomials
• Affine Hecke algebras and raising operators for Macdonald polynomials

Related

• On Combinatorial Formulas for Macdonald Polynomials
• Modified Macdonald polynomials and mu-Mahonian statistics
• Bijections between different combinatorial models for q-Whittaker and modified Hall-Littlewood polynomials
• Macdonald symmetry at q=1 and a new class of inv-preserving bijections on words
• Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials of Types A and C. Extended Abstract