Year: 2026
Type: preprint
Abstract: We establish independence of $q$ as a necessary condition for coefficientwise nonnegativity of normalized principal specialization ratios of monic Macdonald polynomials. For arbitrary pairs of partitions, we characterize independence of $q$. For denominator partitions with largest and smallest parts differing by at least two, every ratio independent of $q$ is a power of $t$. We classify the nonnegative ratios under addition of maximal rows and columns of fixed height. For rows of length at least two, the classification is equivalent to rectangular complementation. For arbitrary initial partitions, the generating polynomials of successive row ratios at the smallest number of variables admissible for all stages have simple negative real zeros with lower bounds for the ratios of consecutive absolute zeros. We derive a reciprocity criterion, bounds for gamma coefficients, and the corresponding Jack limit. For generalized staircases, we determine shortest horizontal and conjugate vertical strip sequences and obtain terminating basic hypergeometric representations.
Source: arXiv (Cornell University)
Author: Shimazaki, Tatsushi
Institution: National Institute of Technology, Akashi College
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213988591