Link patterns and elliptic Hecke algebra

Abstract

We compare three families of geometric objects: Schubert varieties in flag manifolds, matrix Schubert varieties, and Borel orbits of 2-nilpotent matrices. The first family is indexed by permutations, the second by partial permutations, and the third - the most general - by link patterns. Each of the geometric objects mentioned above carries a characteristic class in equivariant elliptic cohomology, defined in the framework provided by Borisov and Libgober. We introduce a Hecke-type algebra that gives inductive formulas for computing the equivariant elliptic classes of link patterns. This requires extending the action of the Hecke algebra to menage partial permutations and link patterns. In studying the extended action, an important role is played by the associated quadratic forms and by the action of reflections on forms. Also, we analyze the specialization of equivariant elliptic classes of link patterns to the corresponding classes of Schubert varieties.