Representations of Iwahori-Hecke algebras

表征理论是纯数学中一个富有成果和有趣的领域，它在量子化学、物理和数论等其他领域有潜在的应用。对称群的表示理论是一个特别丰富的研究领域，它利用了分区的对称性，并将代数和组合学的更广泛领域联系在一起。对于某些正整数n，研究对称群S_n的（不可约）表示的一种具体方法是，通过观察某些域K的代数KS_n上的（简单）模块。在K是特征大于n的代数闭域的情况下，也称为普通表示，S_n的不可约表示非常容易理解，包括它们的度，字符公式等。然而，K的特征小于或等于n的情况，也称为模表示，并没有很好地理解。例如，不可约模表示的维数通常是未知的。Specht模块家族就是一个特别的例子。这些自然定义的模块是对称群的普通表示的所有简单模块。然而，在模块化的情况下，这些Specht模块可能不再是不可约的，因此研究Specht模块的不可约分解成为一个非常有趣的问题。这些被称为“分解数”，研究这些是本研究项目的关键目标。对称群的Iwahori-Hecke代数的表示理论包括S_n的模表示理论作为特例。通过研究更广泛的代数集合的表示，我们对S_n的特殊情况有了更深入的了解。本研究项目的第一部分与Andrew Mathas的著作《对称群的Iwahori-Hecke代数和Schur代数》密切相关。我们特别关注第1、2、3和6章。第一章将对称群作为一个抽象群来研究，提醒读者从群的生成器上的辫状关系中得出的各种性质。然后，作者使用类似于对称群的定义构造了Iwahori-Hecke代数。第二章研究了一类称为“细胞代数”的代数。细胞代数A是一种具有独特的细胞基的代数，它很适合于研究A的表示理论。Iwahori-Hecke代数是细胞代数的一个例子。细胞代数有两个重要的特征。首先，给出a的组成因子与a的单元模同构的滤除a的元胞基；其次，我们可以在每个单元模上定义双线性形式，使我们能够构造所有的简单a模。第三章探讨Iwahori-Hecke代数的模表示理论。我们通过观察tableaux， Specht模块和juys - murphy元素的组合学以及Iwahori-Hecke代数上的不可约模块来研究这些问题。最后，第六章重点讨论了分支规则、正则基和分解矩阵。第六章的最后一节着眼于Ariki-Koike代数，这是Iwahori-Hecke代数的进一步变形。这是进一步研究的潜在途径。总之，我的研究项目的目标是理解对称群的（模）表示理论。我们通过研究更广泛的代数集合上的模块来做到这一点，即Iwahori-Hecke代数，由于是细胞代数，它具有良好的结构。我们使用各种各样的技术，比如tableaux的组合，并通过分解数字来研究普通表示和模块化表示之间的相互作用。通过观察Ariki-Koike代数，也有可能在这个方向上继续研究。

Representation Theory is a fruitful and interesting field of pure mathematics which has potential applications to various other fields such as quantum chemistry, physics, and number theory. The representation theory of the symmetric groups is a particularly rich area of research which utilises the symmetries of partitions, and connects the broader areas of algebra and combinatorics together.One specific way to study the (irreducible) representations of the symmetric groups S_n, for some positive integer n, is by looking at (simple) modules over the algebra KS_n for some field K. In the case where K is an algebraically closed field of characteristic greater than n, also known as the ordinary representations, irreducible representations of S_n are very well understood, including their degrees, character formulae, and more. However, the case where the characteristic of K is less than or equal to n, known also as the modular representations, is not as well understood. For example, the dimensions of irreducible modular representations are not known in general.One particular example is the family of Specht modules. These naturally-defined modules turn out to be all of the simple modules for the ordinary representations of symmetric groups. However, in the modular case, these Specht modules may no longer be irreducible, so studying the decomposition of Specht modules into irreducibles becomes a very interesting question. These are called "decomposition numbers" and studying these is a key goal of this research project.The representation theory of Iwahori-Hecke algebras of the symmetric groups includes the modular representation theory of S_n as a special case. By studying the representations of a broader collection of algebras, we gain more insight into the particular case of S_n.The first part of this research project closely follows the book "Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group" by Andrew Mathas. In particular, we focus on chapters 1, 2, 3, and 6.The first chapter studies the symmetric group as an abstract group, reminding the reader of various properties that follow from the braid relations on the generators of the group. The author then constructs the Iwahori-Hecke algebra, using definitions analogous to those of the symmetric group.The second chapter studies a family of algebras called "cellular algebras". A cellular algebra A is an algebra together with a distinguished cellular basis, which is well adapted to studying the representation theory of A. The Iwahori-Hecke algebras are an example of cellular algebras. There are two important features of cellular algebras. Firstly, the cellular basis which gives a filtration of A with composition factors isomorphic to the cell modules of A. Secondly, we can define bilinear forms on each of the cell modules, which allow us to construct all of the simple A-modules.The third chapter looks at the modular representation theory of the Iwahori-Hecke algebra. We study these by looking at the combinatorics of tableaux, Specht modules and Jucys-Murphy elements, and irreducible modules over the Iwahori-Hecke algebra.Finally, the sixth chapter focuses on branching rules, canonical bases and decomposition matrices. The final section of chapter six looks at Ariki-Koike algebras, a further deformation of particular types of Iwahori-Hecke algebras. This is a potential avenue for further research.In summary, the goal of my research project is to understand the (modular) representation theory of the symmetric groups. We do this by studying modules over a broader collection of algebras, namely the Iwahori-Hecke algebra, which has nice structure due to being a cellular algebra. We use a variety of techniques, such as the combinatorics of tableaux, and look at interaction between ordinary representations and modular representations through decomposition numbers. There is also potential to continue research in this direction by looking at Ariki-Koike algebras.