Graded representations of symmetric groups and related algebras

对称群的表示理论是一个非常活跃的研究分支，它与物理、化学和数学中的许多不同主题都有联系。从某种意义上说，表征理论是对对称性的研究：然而一个群体可以被看作是一个抽象的对称集合，这个群体的表征是一种通过对一个具体物体的作用来实现这些对称性的方式，也就是说，在一个向量空间（比如我们生活的三维空间）上。对称群的表示已经被研究了一个多世纪：这导致了许多强有力的结果，特别是美丽的组合结构。然而，在对称群的模表示的研究中，许多问题仍然没有解决：在这种情况下，我们甚至不知道不可约表示的维数，这些维数是可以用来构造所有表示的构建块。通过研究某些Iwahori-Hecke代数的表示，可以获得所需信息的很大一部分，这是一个更容易处理的问题。Iwahori-Hecke代数的表示理论在其自身的权利是重要的，因为它有许多其他的应用。在过去的20年里，对称群的模表示和所谓的“量子群”之间出现了惊人的联系，量子群最初是为了研究数学物理中的Yang-Baxter方程而定义的。5年前，在Khovanov-Lauda-Rouquier （KLR）代数被发现之后，这些联系被特别精确地发现了。事实证明，我们可以将对称群（或Iwahori-Hecke代数）的表示视为KLR代数的表示。此外，这种观点揭示了以前隐藏的令人兴奋的结构特性：特别是，表征变得分级。该项目的目的是最大限度地利用这一突破性的进步。在项目的第一部分中，将从这些代数提供的新观点来研究有关对称群的某些块和前KLR代数的猜想。第二部分将从KLR代数的角度研究Iwahori-Hecke代数的简单模。第三部分将研究对称群的分级Cartan矩阵的不变量。人们希望量子群和对称群的表示之间的思想可以在两个方向上传递，特别是与对称群相关的组合结构将影响量子群的理论。

Representation theory of symmetric groups is a very active branch of research with connections to physics, chemistry and many different topics across mathematics. In a sense, representation theory is the study of symmetry: whereas a group may be viewed as an abstract set of symmetries, a representation of that group is a way of realising those symmetries through an action on a concrete object, namely, on a vector space (such as the 3-dimensional space we live in). Representations of symmetric groups have been investigated for more than a century: this has led to many strong results and, in particular, to beautiful combinatorial constructions. However, many problems remain unsolved in the study of modular representations of symmetric groups: in this context, we do not even know the dimensions of irreducible representations, which are the building blocks that can be used to construct all representations. A substantial part of the required information can be obtained through the study of representations of certain Iwahori-Hecke algebras, which is a more tractable problem. Representation theory of Iwahori-Hecke algebras is important in its own right, as it has many other applications.In the last 20 years, spectacular connections have emerged between modular representations of symmetric groups and the so-called ``quantum groups'', which were originally defined to study the Yang-Baxter equation in mathematical physics. These connections were made particularly precise 5 years ago, after the discovery of Khovanov-Lauda-Rouquier (KLR) algebras. It turns out that one can view representations of a symmetric group (or an Iwahori-Hecke algebra) as a representation of a KLR algebra. Moreover, this point of view reveals previously hidden exciting structural properties: in particular, the representations become graded. The aim of the project is to exploit this ground-breaking advance to the fullest possible extent. In the first part of the project, conjectures that concern certain blocks of symmetric groups and pre-date KLR algebras will be investigated from the new point of view provided by those algebras. The second part will be devoted to a study of simple modules of Iwahori-Hecke algebras through the lens of KLR algebras. The third part will be an investigation into invariants of graded Cartan matrices of symmetric groups. It is hoped that ideas will be transferred between quantum groups and representations of symmetric groups in both directions, in particular, that combinatorial constructions related to symmetric groups will influence the theory of quantum groups.