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A Game of 1’s and 2’s: Constructing level 2 Demazure Filtrations of level 1 Demazure modules

Jeffrey Wand*, Gonzaga University
Talk Abstract: 
In this talk we dive into the representation theory of Lie algebras. Representation theory is an area of math that studies algebraic structures and the objects they “act on.” In addition, it is also a great tool that takes problems in abstract algebra and turns them into linear algebra problems. More specifically, we will be looking at what is called the current algebra $sl_2 \otimes \mathbb{C}[t]$ , where $sl_2$ is the space of complex $2\times 2$ matrices whose trace is zero, and $\mathbb{C}[t]$ is the space of polynomials with complex coefficients. The combinatorics involved in this talk are motivated by giving an explicit filtration (chain of submodules) of a well-studied family of modules, the level l Demazure modules, which are indexed by a natural number \ell and a ‘vector lambda.’ We know such a construction must exist by Naoi who proved the existence for every $\ell≥1$. In this talk we will construct maps that determines our explicit filtration. In defining these maps we have created a “game” that amounts to turning 1’s into 2’s. This “game” can also be played on a directed graph, turning the filtration question into one that investigates certain ways you can traverse this graph. We will investigate this graph and talk about some generalizations.
Time Slot: 
Saturday, April 2, 2016 - 10:15
Room Number: 
STAG 260