Combinatorial aspects of representation theory and algebraic geometry

点击翻译按钮获取中文摘要

The fundamental approach in algebraic geometry is to study geometrical objects by modelling them using polynomial equations. Using this approach, techniques from algebra can be used to study geometrical objects. Conversely, it is also possible to use geometrical ideas to study algebra. If one has some complicated algebraic object, which could be hard to understand, we can study what are called its "representations". One could compare studying the representations of an algebra to studying the X-rays of a human being. If you have a single X-ray of a person, then you cannot obtain complete information about the person who was X-rayed, but if you have X-rays from a number of different angles, then you can begin to form an accurate impression of the human being. In the same way, if we are interested in an algebra, we study all its representations, or some significant subset of them, to find out more about the algebra. In both algebraic geometry and representation theory, the objects involved can be quite complicated and hard to work with. My plan in this project is to use tools from combinatorics to help analyze these objects.