In 1896 Frobenius showed that many important properties of a finite group could in principle be examined using formulas involving the character ratios of group elements, i.e., the trace of the element acting in a given irreducible representation, divided by the dimension of the representation.

In recent years, the current authors introduced the notion of rank of an irreducible representation of a finite classical group.

One of the motivations for studying rank was to clarify the nature of character ratios for certain elements in these groups.

In fact, two notions of rank were given. The first is the Fourier theoretic based notion of $U$-rank of a representation, which comes up when one looks at its restrictions to certain abelian unipotent subgroups. The second is the more algebraic based notion of tensor rank which comes up naturally when one attempts to equip the representation ring of the group with a grading that reflects the central role played by the few “smallest” possible representations of the group.

Following numerical and theoretical evidences, we conjectured that the two notions of rank agree on a suitable collection called “low rank” representations.

In this note we review the development of the theory of rank for the case of the general linear group $G{L}_n$ over a finite field ${\mathbb{F}}_q$, and give a proof of the “agreement conjecture” that holds true for sufficiently large $q$. Our proof is Fourier theoretic in nature, and uses a certain curious positivity property of the Fourier transform of the set of $m\times n$ matrices over ${\mathbb{F}}_q$ of low enough fixed rank.

In order to make the story we are trying to tell clear, we choose in this note to follow a particular example that shows how one might apply the theory of rank to certain counting problems.

1896 年 Frobenius 表明，原则上可以使用涉及群元素特征比的公式来检验有限群的许多重要性质，即元素在给定不可约表示中的迹除以表示的维数。

近年来，当前作者引入了有限经典群的不可约表示的秩的概念。

研究等级的动机之一是澄清这些组中某些元素的字符比率的性质。

事实上，给出了两个等级的概念。第一个是基于傅立叶理论的概念$你$-表示的等级，当人们查看其对某些阿贝尔单能子群的限制时就会出现。第二个是更基于代数的张量等级概念，当人们试图为群的表示环配备反映群的少数“最小”可能表示所发挥的核心作用的分级时，自然会出现这种概念。

根据数值和理论证据，我们推测秩的两个概念在称为“低秩”表示的合适集合上一致。

在本笔记中，我们回顾了一般线性群情况下秩理论的发展 $G{升}_n$ 在有限域上 ${\mathbb{F}}_q$，并证明“一致猜想”在足够大的情况下成立 $q$. 我们的证明本质上是傅里叶理论，并使用了集合的傅里叶变换的某种奇怪的正性质$米\times n$ 矩阵 ${\mathbb{F}}_q$ 足够低的固定等级。

为了使我们试图讲述的故事更清楚，我们在本笔记中选择遵循一个特定示例，该示例展示了如何将秩理论应用于某些计数问题。