The University of Michigan Combinatorics Seminar


Abstract 

The first part of the talk is concerned with some very explicit
multiplication formulas in the Ktheory of the variety of complete
flags in C^{n}. I start with a Monktype formula, that is, a
formula for the hyperplane section of a Schubert variety in Ktheory.
This formula has been generalized by myself and F. Sottile to a
Pieritype formula in Ktheory. 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 Ktheory 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 Ktheory versions of the Dunkl elements in
the quadratic algebra based on the Monktype 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 Ktheory.
