论文标题
不可约合$ \ text {gl} _n（q）$ - 跨特征中的模块之间的ext组
Ext Groups between Irreducible $\text{GL}_n(q)$-modules in Cross Characteristic
论文作者
论文摘要
令$ g = \ text {gl} _n（q）$为有限字段$ \ mathbb {f} _q $ of $ q $元素的一般线性组，让$ k $是特征性$ r> r> 0 $的代数封闭的字段，这样$ r $ divide $ r $不划分$ q（q-1）$。 1999年，克莱恩（Cline），帕尔沙尔（Parshall）和斯科特（Scott）表明，根据这些假设，$ g $的共同体计算可以将其转换为$ q $ -schur代数的Ext $^i $计算。本文的目的是扩展Cline，Parshall和Scott的结果，并表明$ \ text {gl} _n（q）$的Ext $^i $计算也可以转换为Ext $^i $计算，上面是适当的$ q $ q $ -schur代数（均以$ i = 1 $ i> $ i> 1 $）。为此，我们建立了将$ \ text {gl} _n（q）$的某些ext组与$ q $ -schur algebra $ s_q（n，n，n）_k $的ext组相关的公式。结果，我们表明，没有属于团队主要的Harish-Chandra系列的不可约定的$ kg $模型的自我启动。作为较高程度的应用，我们描述了一种方法，该方法可在不可还原$ kg $模型之间产生较高的ext基团的消失结果，并在一系列示例中演示这种方法。
Let $G=\text{GL}_n(q)$ be the general linear group over the finite field $\mathbb{F}_q$ of $q$ elements, and let $k$ be an algebraically closed field of characteristic $r >0$ such that $r$ does not divide $q(q-1)$. In 1999, Cline, Parshall, and Scott showed that under these assumptions, cohomology calculations for $G$ may be translated to Ext$^i$ calculations over a $q$-Schur algebra. The aim of this paper is to extend the results of Cline, Parshall, and Scott and show that Ext$^i$ calculations for $\text{GL}_n(q)$ may also be translated to Ext$^i$ calculations over an appropriate $q$-Schur algebra (both for $i=1$ and $i>1$). To that end, we establish formulas relating certain Ext groups for $\text{GL}_n(q)$ to Ext groups for the $q$-Schur algebra $S_q(n,n)_k$. As a consequence, we show that there are no non-split self-extensions of irreducible $kG$-modules belonging to the unipotent principal Harish-Chandra series. As an application in higher degree, we describe a method which yields vanishing results for higher Ext groups between irreducible $kG$-modules and demonstrate this method in a series of examples.
