Amanda Lake Heath: On the Minuscule Representation of Types An and Dn
Abstract: In the late nineteenth century, Killing and Cartan discovered and classified all finite-dimensional simple Lie algebras over the complex numbers. Shortly after this, all irreducible representations of these algebras were classified as well. Among these representations, minuscule representations play an important role. It is known that the minuscule representations of simple Lie algebras are irreducible. My goal is to show the irreducibility of a minuscule representation using only the cycle structures of the Weyl group elements viewed as permutations acting on the set of weights of that representation. In this project we focus on the simple Lie algebras of type An and Dn. Using a computer program [Co], we are able to study the cycle structures of these permutations and eliminate possible dimensions of submodules to show the irreducibility of minuscule representations in many cases.