辫群表示的有穷像与KZ型方程的代数解

中文摘要 AI 辅助

有穷单值性在群表示与微分方程的代数解之间架起了一座桥梁。我们研究了Katz-Long-Moody构造中的这一联系，该构造将自由群与辫群的半直积的表示转化为同一群的新表示，并与Knizhnik-Zamolodchikov (KZ)型方程相关。对于固定的有穷像输入，我们分类了使得所得表示在有穷像情况下的参数值，既包括在半直积上的情况，也包括在其自由群和辫群子群上的情况。特别地，辫群像的有穷性与容许参数无关。这些结果给出了相应的正则奇点KZ型方程的所有解为代数的充要条件。在限制到自由群时，它们还刻画了相关Fuchs系统的有穷单值性以及所有解的代数性，将该分类与关于代数超几何函数的经典问题联系起来。

英文摘要

Finite monodromy provides a bridge between group representations and algebraic solutions of differential equations. We study this connection for the Katz-Long-Moody construction, which transforms representations of the semidirect product of a free group and a braid group into new representations of the same group and is related to Knizhnik-Zamolodchikov (KZ)-type equations. For a fixed finite-image input, we classify the parameter values for which the resulting representations have finite image, both on the semidirect product and on its free-group and braid-group subgroups. In particular, finiteness of the braid-group image is independent of the admissible parameter. These results give necessary and sufficient conditions for all solutions of the corresponding regular-singular KZ-type equations to be algebraic. On restriction to the free group, they also characterize finite monodromy and algebraicity of all solutions of the associated Fuchsian systems, connecting the classification to classical questions about algebraic hypergeometric functions.

发表机构

• Chiba University（千叶大学）

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