The Alperin-McKay conjecture for blocks of finite simple groups of Lie type

群形成了对称性的数学语言，在数学和其他科学中得到了应用。一个群的表示允许这个群被实现为矩阵，并应用线性代数的机器。一个基本的任务是描述一个群的不可约表示，所有表示的建筑砖.局部-整体结构，如麦凯猜想，提出不可约对象的数量只取决于局部子群（与群的素数幂结构相关的较小群）中的相似对象.对于一个素数p，a的不可约表示被划分为p-块，每个块都自然地与一个Brauer对应块（局部子群的p-块）相关联。Alperin-McKay猜想是McKay猜想的一个改进，它考虑了p-块的划分。Späth的一个约化定理证明了有限单群表示上的归纳Alperin-McKay条件的成立，并证明了Alperin-McKay猜想的成立.开放的情况下出现有限群的李型和最近的一个新的标准，为这些群体已经提供了由Späth和申请人。拟议的项目旨在建立归纳条件，通过这一新的标准，在所有开放的情况。李型群可以看作是代数群的有限类似物。在Deligne和Lusztig的开创性工作之后，这些群的表示理论已经被深入研究，使用代数几何的机器以及组合工具，如Broué，Malle和Michel建立的d-Harish-Chandra理论。Cabanes和Späth的结果，为了证明McKay猜想，减少了新的标准，考虑一个局部性质，并建立一个合适的bipjet.The局部性质涉及的行动，群自同构的地方表示，这应该遵循明确地建设这些表示，东西，迄今已收到很少的关注。所需的双射将局部和全局高度零表示联系起来，同时还考虑了各种自同构的作用。Späth和申请人已经成功地验证了这一标准在一个更容易的情况下，所使用的方法提出了一个蓝图，以接近其他李类型。希望这个项目也将提供一个更深入的理解，为什么这样的局部-整体猜想成立简单的群体，通过开发一个同时参数化的局部和整体表示的Lie类型的群体。

Groups form the mathematical language for symmetry that is applied within mathematics and other sciences. A representation of a group allows the group to be realised as matrices and apply the machinery from linear algebra. A fundamental task is to describe irreducible representations of a group, the building bricks of all representations.The local-global conjectures, such as the McKay conjecture, propose that the number of irreducible objects depends only on similar objects in local subgroups (smaller groups related to the prime power structures of the group). For a prime number p there is a partition of a groups irreducible representations into p-blocks, each of which is associated naturally with a Brauer correspondent, a p-block of a local subgroup.The Alperin-McKay conjecture is a refinement of the McKay conjecture which takes into account this partition into p-blocks. It postulates that there is a bijection between the height zero irreducible representations of a p-block and those of the Brauer correspondent.A reduction theorem by Späth proves that the Alperin-McKay conjecture would follow from the verification of the inductive Alperin-McKay condition on representations of finite simple groups. The open cases occur as finite groups of Lie type and recently a new criterion tailored to these groups has been provided by Späth and the applicant.The proposed project aims to establish the inductive condition by means of this new criterion in all open cases. Groups of Lie type can be seen as finite analogues of algebraic groups. After the groundbreaking work of Deligne and Lusztig, the representation theory of these groups has been intensively studied using machinery from algebraic geometry together with combinatorial tools such as d-Harish-Chandra theory established by Broué, Malle and Michel. The results of Cabanes and Späth, towards proving the McKay conjecture, reduce the new criterion to considering a local property and establishing a suitable bijection.The local property concerns the action of group automorphisms on local representations, which should follow from explicitly constructing these representations, something that to date has received little attention. The required bijection relates the local and global height zero representations, while also taking into account the action of various automorphisms. Späth and the applicant have been successful in validating this criterion in one of the easier cases and the methods used present a blueprint to approach the other Lie types.Hopefully this project will also provide a deeper understanding of why such a local-global conjecture holds for simple groups by developing a simultaneous parametrisation for the local and global representations of groups of Lie type.