Research Summary and Plan
نویسنده
چکیده
My research focuses on topics related to unipotent classes and Springer theory for algebraic groups. These concepts are intimately related to the representation theory of Weyl groups. I have recently also become interested in complex reflection groups, which can be viewed as generalizations of Weyl groups. Combinatorial methods and algorithms have a long history as a means for studying all of these subjects, and they play a major role in the work that I have done as well. To set the stage for a discussion of this work, we review the basic ideas involved. Let G be a reductive algebraic group over an algebraically closed field k, and let W be its Weyl group. Let U be the variety of unipotent elements in G. By the early work of Jacobson-Morozov, Kostant, and others, we know that the unipotent elements fall into just a finite number of conjugacy classes. Let X denote the set of these conjugacy classes. Now, X = X(G) is equipped with a natural partial order: we say O1 ≤ O2 if O1 ⊂ O2. There is a unique largest unipotent class, whose closure is all of U , as well as a unique smallest one, consisting of just the identity element. Given a class O ∈ X and an element u ∈ O, we can consider the centralizer ZG(u) of u in G, as well as the component group A(u) = ZG(u)/Z G(u). (Of course, choosing a different u in the same O gives an isomorphic centralizer and component group.) The Springer correspondence is a fundamental result relating unipotent classes to representations of W . Specifically, consider the set
منابع مشابه
Preparing effective grant applications.
The abstract and specific aims (described below) sections, two of the most important components of any grant application, must provide a cohesive framework for the application. The abstract provides an outline of the proposed research for the funding agency and the reviewers. Despite the fact that the abstract comes before the research plan in the physical document and is probably read before t...
متن کاملResearch Summary: Datalog-Based Data Access
This paper presents the research summary of a Ph.D. plan concerning the study of Ontology-Based Data Access (OBDA) for ontologies expressed in Datalog-based formalisms, i.e., Datalog rules that allow the use of existential variables in the head.
متن کاملResearch Summary
My main area of research is theoretical computer science, computational complexity and computational logic, with an emphasis on proof complexity and its connection to algebraic complexity. The Summary section below contains a short description of my previous work, my intended research plans and my longterm research goals. The sections after it contain background and a more detailed description ...
متن کاملVisual Indoor Localization with a Floor-Plan Map
In this report, a indoor localization method is presented. The method takes firstperson view video clip as input and uses a 2D floor-plan map as prior knowledge. Motivation, methodology, elementary experimental results, and further extensions are reported and discussed. This report also serves as a brief research summary for CS 6756.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2003