Enumerating classes and characters of p-groups

群是对称的数学语言。每当科学家试图描述对称性时——无论是晶格、某个方程的一组解，还是时空——他们都可能使用群论的语言。有限群论的许多研究都是为了了解一个给定的群有多少种方式可以用矩阵表示。在实践中，对所有这些表示进行分类是一个不可行的难题。我们能不能至少列举出一些有趣群体的代表？在我们的合作项目中，我和新西兰奥克兰的奥布莱恩教授正在研究关于p群的表示和共轭类的枚举问题。如果一个群的大小由质数p的幂给出，那么它就是一个p群。使用p群的一个优点是可以使用一种几何方法来构造所有的表示，称为基里洛夫轨道方法。在以前的工作中，我们已经利用了p群的Kirillov轨道方法，根据有限域上的代数变化，给出了这些群的特征和类的几何描述。在计划的访问期间，我们的目标是完善我们的分析，以回答关于p群的重要具体类别的悬而未决的问题。我们还将在研究出版物中组织我们的结果。

Groups are the mathematical language of symmetry. Whenever scientists aim to describe symmetries - be it of a lattice, the set of solutions to certain equations, or space-time - they are likely to use the language of group theory. Much research in finite group theory is motivated by the desire to understand in how many ways a given group may be represented by matrices. In practice, it is an unfeasibly hard problem to classify all such representations. Can we at least enumerate the representations of interesting groups? In our joint project, Professor O'Brien of Auckland, New Zealand, and I are investigating enumerative questions regarding representations and conjugacy classes of p-groups. A group is a p-group if its size is given by the power of a prime number p. An advantage of working with p-groups is the availability of a geometric method to construct all representations, known as the Kirillov orbit method. In previous work, we have already utilized the Kirillov orbit method for p-groups to give a geometric description of characters and classes of these groups, in terms of algebraic varieties over finite fields. During the projected visit, we are aiming to refine our analysis to answer outstanding questions on important specific classes of p-groups. We will also organize our results in research publications.