正特征下反射群的不变微分导子
Invariant differential derivations for reflection groups in positive characteristic
AI总结：
本文研究了正特征下反射群的不变微分导子理论，给出了特征2下的Saito判据，并获得了SL(n,q)、GL(n,q)及其中间群的显式基。
摘要：
围绕有限反射群的许多迷人数值规律源于 Solomon 1963 年描述不变微分形式的著名定理。在复数域上，不变微分导子也展现出有趣的数值规律。我们探索了任意域上的类似理论，特别是当基础域的特征整除作用反射群的阶且 Solomon 定理的结论可能失效时。利用 Broer 和 Chuai 的结果，我们给出了一个 Saito 判据（雅可比判据），用于寻找在特征 2 域上区分某些情况的有限群不变微分导子基。我们证明了当反射群的横截根空间达到最大时，反射超平面位于单一轨道中，并展示了指数与余指数的对偶性。在这种情况下，一组基本导子被用来通过扭曲楔合构造不变微分导子基。我们获得了特殊线性群 SL(n,q) 和一般线性群 GL(n,q) 以及介于两者之间的所有群的显式基。
英文摘要：
Much of the fascinating numerology surrounding finite reflection groups stems from Solomon's celebrated 1963 theorem describing invariant differential forms. Invariant differential derivations also exhibit interesting numerology over the complex numbers. We explore the analogous theory over arbitrary fields, in particular, when the characteristic of the underlying field divides the order of the acting reflection group and the conclusion of Solomon's Theorem may fail. Using results of Broer and Chuai, we give a Saito criterion (Jacobian criterion) for finding a basis of differential derivations invariant under a finite group that distinguishes certain cases over fields of characteristic 2. We show that the reflecting hyperplanes lie in a single orbit and demonstrate a duality of exponents and coexponents when the transvection root spaces of a reflection group are maximal. A set of basic derivations are used to construct a basis of invariant differential derivations with a twisted wedging in this case. We obtain explicit bases for the special linear groups SL(n,q) and general linear groups GL(n,q), and all groups in between.
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