# Combinatorial proof of a permuted basement Macdonald polynomial identity

## TL;DR

This paper provides a combinatorial proof of a refined identity for permuted basement Macdonald polynomials, extending known symmetry properties and linking them to Kazhdan-Lusztig involution.

## Contribution

It refines the Concha-Lapointe identity for permuted basement Macdonald polynomials and proves it combinatorially, connecting it to Kazhdan-Lusztig involution.

## Key findings

- Refined the Concha-Lapointe identity for a sub-family of permuted basement Macdonald polynomials.
- Provided a combinatorial proof of the refined identity.
- Showed the identity's equivalence to invariance under Kazhdan-Lusztig involution.

## Abstract

A well-known and fundamental property of the Macdonald polynomials $P_\lambda(x;q,t)$ 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.