Representations of cyclotomic Hecke algebras

以一个经典而备受喜爱的数学对象为例，它自19世纪以来一直被广泛研究。我们想看看它行为的一些特殊方面。我们发现有两种情况。在一个环境中，这是很好理解的；然而，在那个环境之外，它在很大程度上是不可预测的。它遵守规则的框架，但除了这些规则强加的限制之外，我们的知识有限。我们的目标是尽可能准确地描述它可以采取的行为方式。但我们的经典朋友只是数学和物理中在不同背景下出现的一类结构中的一个具体例子。尽管大多数成员的行为比原始对象更复杂，但这个类中的所有内容都显示出类似的属性。因此，研究整个班级并适当地专门化我们的结果对我们来说可能是有用的。此外，事实证明，这些新对象与其他看似无关的数学分支之间存在着深刻的联系。经典对象是对称群，我们想知道它的表示。每个表示都像一个乐高结构：它由一组积木组成。构建块是不可约的表示。在复数领域，这些不可约的表示被理解和分类，我们精确地知道如何将它们连接在一起来形成新的表示。因此，在复数上，我们理解对称群的所有表示。但在一个武断的领域，这不再是真的。然而，如果我们知道某些表示的结构，我们就会获得大量的信息，这些表示被称为Speht模块。更一般的对象是G(r，1，n)型割圆Hecke代数，对称群代数是它的特例。当代数不是半单的时候，人们对它的表示知之甚少，尽管它们出现在不同的表示理论环境中，并且已经被证明与代数李理论有着深刻的联系。本研究的主要目的是研究分圆Hecke代数的某些表示的性质。我们想要建立在我们的对称群代数知识的基础上，以找到关于这些表示的信息，但我们也想发展更一般的结果，这将把我们带到新的方向。

Take a classical and much-loved mathematical object, one which has been studied extensively since the 19th century. We want to look at some particular aspects of its behaviour. We find there are two cases. In one environment, it is well understood; however out of that environment, it is, as yet, largely unpredictable. It obeys a framework of rules, but beyond the restrictions imposed by those rules we have limited knowledge. Our aim is to give as precise a description as possible of the ways in which it can behave. But our classical friend is just one specific case in a class of structures that arise in various contexts in mathematics and physics. Everything in this class exhibits similar properties, although most members have more complicated behaviour than the original object. It may therefore be useful to us to study the entire class, and specialize our results as appropriate. Moreover, it turns out that there are deep connections between these new objects and other, seemingly unrelated, branches of mathematics. The classical object is the symmetric group and we want to know about its representations. Each representation is like a Lego structure: it is made up of a set of building blocks. The building blocks are the irreducible representations. Over the field of complex numbers, these irreducible representations are understood and classified, and we know precisely how to join them together to make new representations. Thus over the complex numbers, we understand all representations of the symmetric groups. But over an arbitrary field, this is no longer true. We would, however, gain substantial information if we knew about the structure of certain representations, called Specht modules. The more general objects are the cyclotomic Hecke algebras of type G(r,1,n), of which the symmetric group algebra is a particular case. When the algebra is not semisimple, very little is known about its representations, although they appear in various representation theoretic contexts, and have been shown to have deep links with algebraic Lie theory. The principal aim of this research is to investigate the properties of certain representations of the cyclotomic Hecke algebras. We would like to build on our knowledge of the symmetric group algebras to find information about these representations, but we also want to develop more general results that will take us in new directions.