Topics in the representation theory of finite groups and related algebras

有限群表示论及相关代数的主题

• 批准号: 2114521
• 金额: --
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2114521
• 关键词: Topics; representation; theory; finite; groups

Representation theory is a branch of pure mathematics broadly known as abstract algebra. It is a study of linear group actions, and in particular tries to classify the linear actions in terms of certain basic building blocks known as irreducibles.Classical theory (which goes back over a hundred years to Frobenius and Schur) works over the complex numbers; however modern treatments, and in particular the PhD work of my students work over modular fields of (non-zero) prime characteristic where the prime divides the group order. Taking the group to be the symmetric group (comprising all bijections of a finite set) we can ask a fundamental question: what are the dimensions of the irreducibles over fields of prime characteristic? Our work will use modern techniques from algebra together with some computational techniques in GAP to attack this problem, using related algebraic structures known as Schur algebras. We can also use these methods to consider related questions for more general diagram algebras, such as the Brauer algebra and the partition algebra.

表示论是纯数学的一个分支，广泛称为抽象代数。它是对线性群作用的研究，特别是试图根据某些称为不可约的基本构件对线性作用进行分类。经典理论（可以追溯到一百多年前的 Frobenius 和 Schur）研究复数；然而，现代的治疗方法，特别是我的学生的博士工作，是在（非零）素数特征的模块化域上进行的，其中素数除以群阶。将该群视为对称群（包含有限集的所有双射），我们可以问一个基本问题：素数特征域上不可约的维数是多少？我们的工作将使用代数的现代技术以及 GAP 中的一些计算技术来解决这个问题，并使用称为 Schur 代数的相关代数结构。我们还可以使用这些方法来考虑更一般的图代数的相关问题，例如布劳尔代数和分区代数。

项目成果

Modular Temperley-Lieb Theory

模块化坦珀利-利布理论

• DOI: 10.17863/cam.95095
• 发表时间: 2023
• 作者: Spencer R

Non-induced modular representations of cyclic groups

循环群的非归纳模表示

• DOI: 10.17863/cam.105373
• 发表时间: 2024
• 作者: Spencer R

(l,p)-Jones-Wenzl idempotents

(l,p)-Jones-Wenzl 幂等

• DOI: 10.1016/j.jalgebra.2022.03.022
• 发表时间: 2022
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Martin S