Hunting The Poles in the Staircases

TL;DR

This paper provides a combinatorial explanation for the regularity of certain nonsymmetric Macdonald polynomials at specific specializations, with implications for quantum Hall effect models and algebraic combinatorics.

Contribution

It introduces combinatorial tools and path analysis in the Yang-Baxter graph to explain pole cancellations in Macdonald polynomials at specializations.

Findings

Certain nonsymmetric Macdonald polynomials remain regular at q a t b+1=1

Pole cancellations are explained through symmetries in the Yang-Baxter graph

The approach offers a combinatorial perspective on polynomial regularity

Abstract

Motivated by applications to the fractional quantum Hall effect and, in particular, to the Bernevig-Haldane conjectures, we investigates the behavior of Macdonald polynomials under specializations of the form q a t b = 1. Our main focus is to explain, in a simple and purely combinatorial way, why certain nonsymmetric Macdonald polynomials indexed by staircase vectors with steps of height a and width b remain regular at the specialization q a t b+1 = 1, despite the presence of potential poles in their rational coefficients. To this end, we introduce a set of combinatorial tools that track how poles are created or cancelled along paths in the Yang-Baxter graph. By carefully constructing paths from the zero vector to the staircase and analyzing the resulting denominators, we show that the absence of certain poles follows from intrinsic symmetries and cancellations encoded in the Yang-Baxter graph.