The Combinatorics And Geometry Of The Orbits Of The Symplectic Group On Flags In Complex Affine Space

Abstract

Let F lC2n = B[-] GL2n C be the manifold of flags in C2n . F lC2n has a natural action of S pn by right multiplication. In this thesis we will describe the orbits of S pn on F lC2n . We begin by giving background material in chapter 2 on the combi¨ natorics of S n , the flag manifold, and Grobner bases. In chapter 3 we describe the orbits of B[-] x S pn on full rank 2n x 2n matrices (equivalent to the orbits of S pn on F lC2n ) by mapping those orbits to orbits of B[-] x B+ via M [RIGHTWARDS ARROW] MJM T using [RS90] and then applying the tools available to understand those orbits (see [Ful92]). We recall that the orbits of B[-] x S pn on full rank matrices correspond to fixed-point-free involutions and we explore the combinatorics of the poset of fixed point free involutions to gain insight into the corresponding poset of orbit ¨ closures. We also give a Grobner degeneration of each orbit closure to a union of matrix Schubert varieties. In the chapter 4 we develop understanding of unions of matrix Schubert varieties by finding their equations. In chapter 5 we give the partial results that we have achieved in finding the defining equations for the orbit closures of the orbits of B[-] x S pn .