Year: 2026
Type: preprint
Abstract: The symmetric Macdonald polynomial at peculiar values of parameter $t=q^{-m}$, $m\in\mathbb{Z}_{\ge 0}$ 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.
Source: arXiv (Cornell University)
Authors: Mironov, A., Morozov, A., Popolitov, A.
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Russian Science Foundation
Award: G7772039525
OpenAlex ID: W7206197782