关于Donkin的倾斜模猜想IV：新前沿

中文摘要 AI 辅助

设$G$是定义在$\mathbb{F}_{p}$上的单连通简单代数群概形，$G_{r}$为其第$r$个Frobenius核。Donkin著名的倾斜模猜想声称，给定的不可分解内射$G_{r}$-模可以实现为某个特定倾斜模的限制。该猜想于1990年首次提出，近30年来一直未被证明。作者于2019年发现了一个反例。本文提供了围绕倾斜模猜想的一些丰富数学的指示，并讨论了一个由Humphreys和Verma提出的更早的猜想。基于多个新反例族，本文提出了倾斜模猜想的新版本。最后，通过一项新的计算，作者提供了进一步证据，表明对于底层根系为$\mathrm{A}_{n}$型的约化代数群，在所有特征$p > 0$的域上，倾斜模猜想应当成立。

英文摘要

Let $G$ be a simple, simply connected algebraic group scheme defined over $\mathbb{F}_{p}$, and let $G_{r}$ be the $r$th Frobenius kernel. Donkin's famous Tilting Module Conjecture purports that a given indecomposable injective $G_{r}$-module can be realized as the restriction of a specific tilting module. The conjecture was first stated in 1990 and withstood proof for nearly 30 years. The authors discovered a counterexample in 2019. This paper provides some indication about the rich mathematics surrounding the Tilting Module Conjecture that also discusses an older conjecture by Humphreys and Verma. New versions of the Tilting Module Conjectures are formulated given multiple families of new counterexamples. Finally, through a new calculation, the authors present further evidence that the Tilting Module Conjecture should hold for reductive algebraic groups with an underlying root system of Type $\mathrm{A}_{n}$ for all fields of characteristic $p > 0$.

发表机构

• University of Wisconsin-Stout（威斯康星大学斯托特分校）
• University of Georgia（佐治亚大学）
• University of South Alabama（南阿拉巴马大学）
• Georgia Southern University（佐治亚南方大学）

机构由 AI 辅助整理，请以论文原文为准。