Macdonald principal specialization ratios and their generating polynomials
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.
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BibTeX
@misc{shimazaki2026macdonald,
title = {{Macdonald principal specialization ratios and their generating polynomials}},
author = {Tatsushi Shimazaki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.21431},
eprint = {2609.21431},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21431},
}
RIS
TY - UNPB
TI - Macdonald principal specialization ratios and their generating polynomials
AU - Tatsushi Shimazaki
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.21431
UR - https://arxiv.org/abs/2609.21431
AB - 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.
ER -