Combinatorial proof of a permuted basement Macdonald polynomial identity

2025/08/28 by Orr, Daniel, Rivera, Johnny
#Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2508.20337

Abstract

A well-known and fundamental property of the Macdonald polynomials P_λ(x;q,t) is their invariance under the transformation sending (q,t) to (q^-1,t^-1). Recently, Concha and Lapointe showed that this property extends in an interesting, nontrivial way to an identity for partially symmetric Macdonald polynomials. Their identity played a key role in the work of Bechtloff Weising and Orr linking partially symmetric Macdonald polynomials to parabolic flag Hilbert schemes. In this paper, we refine the Concha-Lapointe identity to a sub-family of Alexandersson's permuted basement Macdonald polynomials and give a combinatorial proof of the refined identity. We show also that the Concha-Lapointe identity is equivalent to the assertion that (normalized) partially symmetric Macdonald polynomials are fixed under the Kazhdan-Lusztig involution.

Citations

• Flagged LLT polynomials, nonsymmetric plethysm, and nonsymmetric Macdonald polynomials
• Stable-limit partially symmetric Macdonald functions and parabolic flag Hilbert schemes
• A Geometric Realization of Partially-Symmetric Macdonald Polynomials
• Type A Partially-Symmetric Macdonald Polynomials
• The \mathbbA_q,t algebra and parabolic flag Hilbert schemes
• Standard bases for affine parabolic modules and nonsymmetric Macdonald polynomials
• Hilbert schemes, polygraphs, and the Macdonald positivity conjecture

Related

• A Geometric Realization of Partially-Symmetric Macdonald Polynomials
• Type A Partially-Symmetric Macdonald Polynomials
• Some remarkable new Plethystic Operators in the Theory of Macdonald Polynomials
• Macdonald polynomials and symmetric functions
• A combinatorial formula for Sahi, Stokman, and Venkateswaran's generalization of Macdonald polynomials