GLₙ(O₂)共轭类的典范截面方法
A canonical section method for conjugacy classes of GL_n(O_2)
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中文总结 AI 辅助
本文提出GLₙ(O₂)共轭类的典范截面方法，证明其类方程仅依赖剩余域基数，该论证早于并独立于后续相关分类方法，因具独立意义而见刊。
中文摘要 AI 辅助
设O为具有有限剩余域的非阿基米德局部域的整数环，p为其极大理想，O₂=O/p²。我们证明GLₙ(O₂)的类方程仅通过剩余域的基数依赖于O：对于两个具有同构有限剩余域的此类环O和O'，存在GLₙ(O₂)与GLₙ(O'₂)的共轭类之间的典范双射，且保持每个类的大小。该论证构造了归约映射GLₙ(O₂)→GLₙ(O₁)的一个截面，该截面在其块若尔当典范形中任意元素的中心化子上是乘法的，并利用它将位于GLₙ(O₁)固定类之上的共轭类的分类从一个环转移到另一个环。本笔记记录了作者2010年博士论文中针对该结果的原始论证，该论证早于且独立于两个后续的密切相关陈述的证明：Prasad、Singla和Spallone针对n≤4的相似类的扩张（Ext）理论分类，以及Jambor和Plesken针对长度为2的一般单链环的同态（Hom）理论方法。我们在此记录中心化子-截面论证，因为它似乎具有独立意义，且多位同事要求将其见刊。
英文摘要
Let O be the ring of integers of a non-Archimedean local field with finite residue field, let p be its maximal ideal, and let O_2 = O/p^2. We show that the class equation of GL_n(O_2) depends on O only through the cardinality of its residue field: for two such rings O and O' with isomorphic finite residue fields, there is a canonical bijection between the conjugacy classes of GL_n(O_2) and of GL_n(O'_2) which preserves the size of every class. The argument constructs a section of the reduction map GL_n(O_2) -> GL_n(O_1) which is multiplicative on the centralizer of any element in its block Jordan canonical form, and uses it to transport the classification of conjugacy classes lying above a fixed class of GL_n(O_1) from one ring to the other. This note records the original argument for this result, developed in the author's 2010 doctoral thesis, which predates and is independent of two later proofs of closely related statements: the Ext-theoretic classification of similarity classes for n <= 4 by Prasad, Singla and Spallone, and the Hom-theoretic approach of Jambor and Plesken for general uniserial rings of length two. We record the centralizer-section argument here since it seems to be of independent interest and several colleagues have asked to see it in print.
