Representations of Iwahori-Hecke algebras

表示论是一个卓有成效和有趣的纯数学领域，它在量子化学、物理学和数论等其他领域都有潜在的应用。对称群的表示理论是一个特别丰富的研究领域，它利用了划分的对称性，并将代数和组合学的更广泛的领域联系在一起。研究对称群S_n的(不可约)表示的一个具体方法是，对于某个正整数n，通过查看域K的代数Ks_n上的(简单)模，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的不可约表示非常好地理解，包括它们的次数、特征标公式等。然而，K的特征小于或等于n的情况，也称为模表示，并不是很好地理解。例如，不可约模表示的维度一般是未知的。一个特别的例子是specht模族。这些自然定义的模被证明是对称群的普通表示的所有简单模。然而，在模的情况下，这些specht模可能不再是不可约的，所以研究specht模到不可约的分解成为一个非常有趣的问题。对称群的Iwahori-Hecke代数的表示理论包括了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。本研究的第一部分紧跟着安德鲁·马萨斯的著作《对称群的Iwahori-Hecke代数和Schur代数》。第一章把对称群作为一个抽象的群来研究，提醒读者从群的生成元上的辫子关系得到的各种性质。然后，作者利用类似于对称群的定义，构造了Iwahori-Hecke代数。第二章研究了一族称为“元胞代数”的代数。胞代数A是一个具有特殊胞基的代数，它非常适合于研究A的表示理论。Iwahori-Hecke代数是胞代数的一个例子。元胞代数有两个重要的特征。首先，给出A的一个具有同构合成因子的滤子的胞基；其次，我们可以在每个胞模上定义双线性形式，这使得我们可以构造所有的单A-模。第三章讨论Iwahori-Hecke代数的模表示理论。我们通过研究Iwahori-Hecke代数上的Tableaux、specht模和Jucys-Murphy元以及不可约模的组合来研究这些问题。最后，第六章讨论了分支规则、标准基和分解矩阵。第六章的最后一节讨论了Ariki-Koike代数，它是特殊类型的Iwahori-Hecke代数的进一步变形。总之，我的研究项目的目的是了解对称群的(模)表示理论。我们通过研究更广泛的代数集合上的模来做到这一点，即Iwahori-Hecke代数，它由于是细胞代数而具有良好的结构。我们使用各种技术，例如表的组合学，并通过分解数来查看普通表示和模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.