The University of Michigan Combinatorics Seminar
Winter 2003
April 4, 4:10-5:00, 3866 East Hall





The K-theory of the flag variety and the Fomin-Kirillov quadratic algebra

Cristian Lenart

SUNY - Albany




Abstract

The first part of the talk is concerned with some very explicit multiplication formulas in the K-theory of the variety of complete flags in Cn. I start with a Monk-type formula, that is, a formula for the hyperplane section of a Schubert variety in K-theory. This formula has been generalized by myself and F. Sottile to a Pieri-type formula in K-theory. Both formulas are expressed in terms of chains in the Bruhat order on the symmetric group. In a joint project with A. Yong, we attempt to generalize to K-theory the construction of the cohomology of the flag variety as a commutative subalgebra of a certain associative algebra with degree 2 relations; this construction is due to S. Fomin and A. Kirillov. In order to carry out our generalization, we define K-theory versions of the Dunkl elements in the quadratic algebra based on the Monk-type formula, and show that they commute. We state a conjecture leading to the mentioned construction, and discuss the stage of its proof. Finally, we discuss the implications of this construction to finding a combinatorial rule for multiplying Schubert classes in K-theory.