Anna Bertiger

Ph.D. (2013) Cornell University

First Position

University of Waterloo, Postdoctoral Fellow


The Combinatorics and Geometry of the Orbits of the Symplectic Group on Flags in Complex Affine Space


Research Area

representation theory, algebraic geometry, and combinatorics


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 .