Representation Theory of the Partition Algebras and Symmetric Groups

背景：对称群的表示现代科学中最重要的概念工具之一是对称性的概念。数学家赫尔曼·魏尔（Hermann Weyl）给出了一个精确的定义：对称性是物体（例如空间中的粒子配置）在某些变换下的不变性。保持一个对象不变的所有变换的集合，形成所谓的组。表示论是研究抽象群（或代数）如何变换空间的理论。对称群是最重要的群族之一，因为任何有限群都可以在某个对称群中找到。在复数的代表性的对称群进行了分类和建设的青年在1930年。然而，在具有正特征的域（所谓的模情形）上，情况要复杂得多。不可约表示的分类（即粘在一起形成任何表示的构建块）存在，但构建这些表示并理解它们如何粘在一起仍然是这一领域的主要开放问题。一个非常有益的做法一直是有关的代表性的对称群的其他代数对象承认更丰富的数学结构。经典的例子是通过Schur-Weyl对偶与一般线性群的关系，它允许将一些工具从李理论转移到对称群方面。最近，Jones [11]和Martin [12]证明了在复数上，对称群和另一个最初出现在数学物理中的代数结构（即划分代数）之间存在类似的对偶。虽然这种对偶性是在20世纪90年代建立的，但令人惊讶的是，从那时起，很少有人做工作来发展和利用这种联系，以促进我们对对称群的表示论的理解，特别是在模的情况下。人们普遍认为，琼斯和马丁建立的对偶性将延续到任意领域，尽管据我所知，还没有可用的证据。因此，我的第一个目标是将对称群和划分代数之间的对偶性扩展到具有正特征的领域。我的第二个目标是系统地研究由这种对偶性产生的两个模范畴之间的函子。我期望以这种方式获得对对称群的模表示论的实质性的新见解。应该指出的是，许多开放的问题也仍然存在于划分代数的模表示理论。我的第三个目标是通过构建一个“仿射版本”的分区代数来解决这些问题。这是受到类似的结构的对称群和布劳尔代数带来了非常重要的见解，他们的代表性理论。方法和工作计划为了达到我的第一个目标（对偶的扩展到模的情况下），我计划使用类似的方法，在不变量理论[6]和Brauer代数[7]。研究由这种对偶性产生的所谓Schur函子（第二个目标）很可能需要为划分代数引入新的模类，例如广义置换和Young模（参见[10] Brauer代数的类似结果）。我还计划在经典Schur-Weyl对偶中使用[9]中发展的一些方法。然而，应该注意的是，我们的新环境与布劳尔或经典环境有一个主要区别，那就是我们没有一个李论对象在起作用。我期望第三个目标（仿射划分代数的构造）将在这种情况下提供必要的额外结构。

BACKGROUND: REPRESENTATIONS OF THE SYMMETRIC GROUPSOne of the most important conceptual tools in the modern sciences is the notion of symmetry. A precise definition was given by the mathematician Hermann Weyl: symmetry is the invariance of an object ( e.g. a configuration of particles in space) under certain transformations. The set of all transformations leaving an object unchanged form what is called a group. Representation theory is the study of how an abstract group ( or an algebra) can transform a space. The symmetric groups form one of the most important families of groups, as any finite group can be found inside some symmetric group. Over the complex numbers the representations of the symmetric groups were classified and constructed by Young in 1930. However, over fields of positive characteristic ( the so-called modular case) the situation is considerably more complicated. A classification of the irreducible representations (that is the building blocks which, glued together, form any representation) exists but constructing these and understanding how they can be glued together are still major open problems in this area. A very profitable approach has been to relate the representations of the symmetric groups to that of other algebraic objects which admit richer mathematical structures. The classical example is the relationship with the general linear group via Schur-Weyl duality, which allows to transfer some of the tools from Lie theory to the symmetric group side. More recently, Jones [ll] and Martin [12] proved the existence of a similar duality, over the complex numbers, between the symmetric group and another algebraic structure which originally arose in Mathematical Physics, namely the partition algebra. Although this duality was established in the 1990's, surprisingly little work has been done since to develop and exploit this connection to advance our understanding of the representation theory of the symmetric group, especially in the modular case. MAIN OBJECTIVES OF THE PROPOSED RESEARCHIt is widely expected that the duality established by Jones and Martin will carry over to arbitrary fields, although as far as I am aware there is no available proof yet. My first objective is therefore to extend the duality between the symmetric group and the partition algebra to fields of positive characteristics. My second objective will then be to develop a systematic study of the functors between the two module categories which arise from this duality. I expect to obtain substantial new insight into the modular representation theory of the symmetric group in this way. It should be noted that many open problems also remain in the modular representation theory of the partition algebra. My third objective aims to address these by constructing an 'affine version' of the partition algebra. This is inspired by similar constructions for the symmetric group and the Brauer algebra which have brought very significant insights into their representation theory. METHODOLOGY AND WORK PLANTo reach my first objective (the extension of the duality to the modular case) I plan to use methods similar to those developed in invariant theory in [6] and for the Brauer algebra in [7]. The study of the so-called Schur functors arising from this duality (second objective) will most likely require the introduction of new classes of modules for the partition algebra, such as generalised permutation and Young modules (see [10] for analoguous results for the Brauer algebras). I also plan to use some of the methods developed in [9] in the classical Schur-Weyl duality. It should be noted, however, that our new setting has a major difference from the Brauer or the classical context in that we not have a Lie theory object in play. I expect that the third objective (the construction of an affine partition algebra), will provide the necessary extra structure in this case.