Reflection Group Combinatorics

本文的目的是研究有限反射群组合中的各种问题，在不同的一般性水平上。在代数组合学这门学科中，有相当一部分的努力是致力于寻找定义各种组合对象的适当的通用性。反思组通常是找到“正确的”一般性的关键。在项目中出现的一些群是——有限一般线性群，——正多面体的对称群（包括经典的实多面体，以及由G.C. Shephard和H.S.M. Coxeter考虑的复正多面体），以及——代数群和李代数理论中出现的Weyl群。研究这些对称群的一个原因是它们的透明之美，古希腊人已经知道了这一点，即五个正立方体的对称性：立方体、四面体、八面体、十二面体和二十面体。与此同时，这种美也让我们感到沮丧。人们想要理解这些对象，而不是把它们当作五件事清单上不相关的项目：人们想要理解它们的共同特征，以及如何在统一的原则下同时理解它们。这是这个项目的主要目标之一。

This proposal is to investigate various question in the combinatorics of finite reflection groups, at different levels of generality. A significant fraction of effort in the subject of algebraic combinatorics is devoted toward finding the right level of generality to define various combinatorial objects. Reflection groups are often the key to finding the "correct" generality. Some of the groups appearing within the projects are -- the finite general linear groups, -- the groups of symmetry of regular polytopes (both the classical real ones, as well as the complex regular polytopes considered by G.C. Shephard and by H.S.M. Coxeter), and -- the Weyl groups, arising in the theory of algebraic groups and Lie algebras.One reason to study these symmetry group comes from their transparent beauty, known in part already to the ancient Greeks, as the symmetries of the five regular solids: the cube, tetrahedron, octahedron, dodecahedron, and icosahedron. At the same time, this beauty frustrates us. One wants to comprehend these objects not as unrelated items on a list of five things: one wants to understand the features they have in common, and how they can be understand all at once, in terms of unified principles. This is one of the main goals of this project.