Generating equivalence classes of B -stable ideals
This thesis deals with structures found in semisimple Lie algebras called B-stable ideals. We develop a series of simple "moves", ways to move from one B-stable ideal to a smaller one, that preserve certain data arising from each ideal. Specifically, each ideal corresponds to a nilpotent orbit in the Lie algebra and we present a set of moves allowing us to create equivalence classes of ideals corresponding to the same orbit. Additionally, each ideal corresponds to a subgroup of the component group associated to its orbit and we similarly use a set of moves to look at equivalence classes of ideals corresponding to conjugate subgroups. Finally, we analyze these equivalence classes in light of some previously known geometric results, leading us to information about the Springer fiber associated to a given orbit.