AI 中文总结
该研究揭示傅里叶变换对特定不规则联络斯托克斯数据的作用由组合结构支配，通过盖尔变换关联隐性与次优解，建立斯托克斯表示与弗里兹模式的对应并给出闭式公式。
AI 中文摘要
我们研究了复仿射直线上带有无穷远对称不规则类的不规则联络的斯托克斯数据上傅里叶变换的作用，分别从斯托克斯过滤局部系统和斯托克斯局部系统的角度进行分析，证明其受丰富的组合结构支配：（1）观察到在此设定下，斯托克斯过滤完全由其隐性或次优解空间的数据确定，并结合T. Mochizuki的结果，我们证明傅里叶变换通过射影空间中点配置的盖尔变换实现隐性解与次优解的交换；（2）我们证明隐性解与斯托克斯局部系统的等价性与Morier-Genoud-Ovsienko-Schwartz-Tabachnikov得到的点配置、超周期线性差分方程和弗里兹模式之间的三重性密切相关：除符号外，差分方程和弗里兹的系数与非平凡斯托克斯矩阵元素一致。从该斯托克斯-弗里兹对应可知，斯托克斯表示的傅里叶变换由其组合盖尔变换给出，进而得到显式闭式公式。
英文摘要
We study the action of the Fourier transform on the Stokes data of irregular connections on the complex affine line with symmetric irregular classes at infinity, both from the point of view of Stokes filtered local systems and of Stokes local systems, and we show that it is governed by a rich combinatorial structure: (1) Observing that, in this setup, a Stokes filtration is fully determined by the data of either its recessive or subdominant solution spaces, and making the link with results of T. Mochizuki, we show that the Fourier transform amounts to exchanging recessive and subdominant solutions via the Gale transform of configurations of points in projective spaces. (2) We show that the equivalence between recessive solutions and Stokes local systems is deeply connected with the triality relating point configurations, superperiodic linear difference equations and frieze patterns obtained by Morier-Genoud-Ovsienko-Schwartz-Tabachnikov: Up to signs, the coefficients of the difference equations and friezes coincide with the nontrivial Stokes matrix entries. It follows from this Stokes-frieze correspondence that the Fourier transform of Stokes representations is given by their combinatorial Gale transform, leading to explicit closed formulas.
Comments59 pages, 19 figures
