Non-symmetric triads and Baker-Akhiezer functions
Abstract
The symmetric Macdonald polynomial at peculiar values of parameter t=q−m, m∈ℤ≥ 0 is naturally split into non-symmetric parts, which are the (quasi)polynomial Baker-Akhiezer (BA) functions. One may think this is due to symmetricity, and one just picks up this way non-symmetric parts already containing all the information. However, we demonstrate that, in the case of {\bf non-symmetric} Macdonald polynomials, it still works, though each single BA function splits into N! distinct (quasi)polynomial BA functions. The sum of these functions gives rise to the universal solution of the eigenstate problem for the Cherednik Hamiltonians. Extending to arbitrary values of t is also immediate giving rise to counterparts of the Noumi-Shiraishi power series. Altogether, this power series and its reductions to non-symmetric Macdonald polynomials and to BA functions form a non-symmetric triad. There are N! different branches of the non-symmetric triad, each branch being split into N! distinct triads, and of these (N!)2 triads N!(N−1)! are independent. We describe in detail the simplest N=2 case.
Related
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BibTeX
@misc{mironov2026nonsymmetric,
title = {{Non-symmetric triads and Baker-Akhiezer functions}},
author = {A. Mironov and A. Morozov and A. Popolitov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01517},
eprint = {2609.01517},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01517},
}
RIS
TY - UNPB
TI - Non-symmetric triads and Baker-Akhiezer functions
AU - A. Mironov
AU - A. Morozov
AU - A. Popolitov
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01517
UR - https://arxiv.org/abs/2609.01517
AB - The symmetric Macdonald polynomial at peculiar values of parameter t=q^−m, m∈ℤ_≥ 0 is naturally split into non-symmetric parts, which are the (quasi)polynomial Baker-Akhiezer (BA) functions. One may think this is due to symmetricity, and one just picks up this way non-symmetric parts already containing all the information. However, we demonstrate that, in the case of {\bf non-symmetric} Macdonald polynomials, it still works, though each single BA function splits into N! distinct (quasi)polynomial BA functions. The sum of these functions gives rise to the universal solution of the eigenstate problem for the Cherednik Hamiltonians. Extending to arbitrary values of t is also immediate giving rise to counterparts of the Noumi-Shiraishi power series. Altogether, this power series and its reductions to non-symmetric Macdonald polynomials and to BA functions form a non-symmetric triad. There are N! different branches of the non-symmetric triad, each branch being split into N! distinct triads, and of these (N!)^2 triads N!(N−1)! are independent. We describe in detail the simplest N=2 case.
ER -