Cohen-Lenstra heuristics, and ordinary representations of finite groups

理解和测量对称性是纯数学中的一个基本问题。传统上，“对称”一词适用于几何形状，例如，指规则多边形或实体的旋转和反射。然而，在19世纪伊瓦斯特·伽罗瓦的开创性贡献之后，我们已经学会了在更广泛的意义上理解对称，对称的概念已经被群论和后来的表象理论建立在强有力的严格基础上。如今，我们通过群体行动的语言来表达对称的概念。在纯数学中，两个最基本的群作用是集合上的作用(“G-集合”)和向量空间上的作用(“线性表示”)。理解将一个具有群作用的集合转化为具有诱导群作用的向量空间的自然过程是一个古老的问题，在代数内外有许多应用，并且有丰富的文献。在之前与Tim Dokitser的合作中，我们已经完全理解了这一过程的一个方面，即不同的G-集产生相同的表示，从而解决了一个超过60年的问题。我们在G-集上可以得到哪些表示的对偶问题上也取得了相当大的进展。在这个项目中，我建议用几个更重要的有限群无穷族来解决后一个问题的例子。数学中最古老的分支是被称为数论的领域，今天最大的公开问题可以追溯到古希腊人。拟议项目的第二部分将与亨德里克·伦斯特拉共同实施，处于表象理论和数论的交汇点。其目的是研究几个经典数论不变量的对称群，如类群。高斯是第一个提出有关班级群体结构的统计问题的人，例如，他们多长时间是微不足道的，他们的规模在家庭中增长得有多快。其中许多问题至今仍悬而未决。但我们在这一领域的概念理解被科恩和伦斯特拉在80年代初的一篇论文彻底改变了，他们提出，自然界中代数对象出现的频率的主要因素是这个对象的对称性的数量(高阶的是它的自同构群的大小)。他们的启发式工作“开箱即用”，并与在最简单和研究最多的理想类群族(那些虚构的二次场)中的数值实验非常吻合，但它对任意族的推广似乎偏离了基本思想，并以特别的方式修改了假设的概率权重。到目前为止，对这些修改的概念解释仍然难以捉摸。在这个项目中，我将开发一个框架，允许比较自同构群的大小，即使这些群是无限的。这将允许以更概念化的方式重塑最初的科恩-列斯特拉启发式，以更概念化的方式适用于一般家庭，但它也将使其适用于更一般的情况。我计划使用这个框架来研究许多重要的数论不变量的其他统计性质，例如类群，所谓的K-群，以及椭圆曲线的塞尔默群。这些都是数论中最迷人、最神秘的物体。为此，我将开发的代数机器也将具有内在的兴趣，并将应用于其他领域的分布问题，例如几何(双曲流形的同调)和组合学(图的雅可比)。

One of the fundamental problems in Pure Mathematics is to understand and measure symmetries. Classically, the word "symmetry" was applied to geometric shapes, e.g. referring to rotations and reflections of regular polygons or solids. However, after the ground breaking contributions of Évariste Galois in the 19th century, we have learned to understand symmetries in a much wider sense, and the notion of symmetry has been put on a powerful rigorous footing by group theory, and later by representation theory. These days, we express the idea of symmetry through the language of group actions. Two of the most fundamental group actions in pure mathematics are actions on sets ("G-sets"), and actions on vector spaces ("linear representations"). It is an old problem with many applications in and outside of algebra, and with a rich literature, to understand the natural procedure that turns a set with a group action into a vector space with the induced group action. In previous joint work with Tim Dokchitser, we have completely understood one side of this procedure, namely when distinct G-sets give rise to the same representation, thereby settling an over 60 year old problem. We have also made considerable progress on the dual question of which representations can be obtained from G-sets. In this project, I propose to settle instances of this latter problem for several further important infinite families of finite groups.The oldest branch of mathematics is the area called number theory, the biggest open problems today going back to the ancient Greeks. The second part of the proposed project, to be carried out jointly with Hendrik Lenstra, lives at the intersection of representation theory and number theory. The aim is to study symmetry groups of several classical number theoretic invariants, such as class groups. Gauss was the first to ask statistical questions about the structure of class groups, e.g. how often are they trivial, and how fast does their size grow in families. Many of these questions are open to this day. But our conceptual understanding in this area was revolutionised by a paper of Cohen and Lenstra from the early 80s, who proposed that the main factor that accounts for the frequency of algebraic objects in nature is the number of symmetries of this object (in high-browese the size of its automorphism group). Their heuristic works "out of the box" and agrees very well with numerical experiments in the easiest and most-studied family of ideal class groups (those of imaginary quadratic fields), but its generalisations to arbitrary families seem to deviate from the basic idea and to modify the postulated probability weights in ad-hoc ways. Until now, a conceptual explanation of these modifications has remained elusive. In this project, I will develop a framework that allows to compare sizes of automorphism groups, even when those groups are infinite. This will allow to recast the original Cohen-Lenstra heuristic for general families of class groups in a much more conceptual way, but it will also make it applicable in many more general situations. I plan to use this framework to investigate other statistical properties of many important number theoretic invariants, such as class groups, so-called K-groups, and also Selmer groups of elliptic curves. Those are some of the most fascinating and mysterious objects in number theory. The algebraic machine that I will develop to this end will also be of intrinsic interest, and will have applications to distribution questions in other areas, e.g. in geometry (to homology of hyperbolic manifolds) and combinatorics (to Jacobians of graphs).