AI 中文总结
本文研究非对称Macdonald多项式与Baker-Akhiezer函数，发现其BA函数会分解为N!个不同BA函数，构成非对称三元组，还给出N=2的具体情形，为Cherednik哈密顿量本征态问题提供通解。
AI 中文摘要
在参数t=q^(-m)（m为非负整数）的特殊值下，对称Macdonald多项式自然分解为非对称部分，即（拟）多项式Baker-Akhiezer（BA）函数。有人可能认为这源于对称性，仅通过这种方式就能提取包含所有信息的非对称部分，但我们证明，对于非对称Macdonald多项式，该规律依然成立，只是单个BA函数会分解为N!个不同的（拟）多项式BA函数。这些函数的和构成了切列德尼克（Cherednik）哈密顿量本征态问题的通解，推广到t的任意值也很直接，进而产生Noumi-Shiraishi幂级数的对应物。总体而言，该幂级数及其向非对称Macdonald多项式和BA函数的约化形式构成一个非对称三元组。非对称三元组有N!个不同分支，每个分支又分解为N!个不同的三元组，在这(N!)²个三元组中，N!(N-1)!个是独立的。我们详细描述了最简单的N=2情形。
英文摘要
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.
Comments19 pages
