IRFP: The Geometry and Combinatorics of Symmetric Subgroup Orbit Closures on Flag Varieties

Wyser Benjamin J, Athens GA

Investigators

Abstract

The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad. This award will support a twenty-four-month research fellowship by Dr. Benjamin Wyser to work with Dr. Michel Brion at Universite Joseph Fourier Mathematics in Grenoble, France. Given a complex reductive group G and an involution of G, one can consider the subgroup K of G consisting of elements fixed by the involution. Such a subgroup is referred to as "symmetric". The set of K-orbits on the flag variety for G is finite, and comes with a natural partial order. The poset structure is described by elaborate combinatorial machinery. If this combinatorial structure is understood, then information about one orbit closure can often allow one to draw a conclusion about another orbit closure related to it in the partial order. To give an example, given a formula for the equivariant fundamental cohomology class of one orbit closure, a formula for the class of a higher orbit closure can be computed from it using something known as a divided difference operator. One goal of this project is to determine the extent to which this computation can be extended from equivariant cohomology to equivariant K-theory. Using techniques of localization, it should be possible to compute formulas for the equivariant K-theory classes of closed orbits. The goal is to then use those formulas as a basis for a recursion by which divided difference operators would give the remaining formulas. This approach is known to work for Schubert varieties, for instance. A certain class of K-orbit closures (called "multiplicity-free") should be amenable to these techniques as well, since their singularities are known to be no worse than those of Schubert varieties. However, not all examples are multiplicity-free. Indeed, there exist some symmetric subgroups whose orbit closures do exhibit worse singularities. For such orbit closures, new ideas may be needed to compute their K-theory classes, perhaps involving a more detailed study of the exact nature of their singularities. Another main goal is to investigate possible further roles of K-orbit closures in positive Schubert calculus on the full flag variety, building upon recent results in this area. The main idea is to compare a recently discovered rule, which relates some Schubert structure constants to chains in a poset of K-orbits, to other known rules in instances where the cases handled by various rules overlap. The hope is that in examining how the various rules agree in such cases, one rule which generalizes all of them can be found. Schubert varieties are objects which are of great importance in many areas of mathematics, such as algebraic geometry, combinatorics, and representation theory. They have been studied exhaustively by experts in all of these fields, and many deep and interesting connections between them have been discovered as a result. K-orbits and their closures have much in common with Schubert varieties. However, they are not nearly as well-studied, particularly from the geometric and combinatorial viewpoints. Recent results suggest that these objects are worthy of further attention. They hold the potential to reveal yet more of the deep connections between these various interrelated areas of mathematics.