Orbit closures of type A quiver representations are algebraic varieties that arise naturally in several areas of mathematics: for example, in Lusztig’s geometric realization of Ringel’s work on quantum groups; as generalizations of determinantal varieties in commutative algebra; and in the theory of degeneracy loci of maps of vector bundles.
For equioriented type A quivers, a formula due to Knutson-Miller-Shimozono expresses the equivariant cohomology class of each orbit closure as a sum, over certain ‘lacing diagrams’, of products of Schubert polynomials. Lacing diagrams were introduced by Abeasis and del Fra in 1982 to visualize direct sum decompositions of type A quiver representations.
In joint work with Allen Knutson and Jenna Rajchgot, we proved a 2004 conjecture of Buch and Rimnyi that generalizes this formula in two ways: to arbitrarily oriented type A quivers, and to equivariant K-classes (a.k.a. K-polynomials), from which equivariant cohomology can be recovered.
The aim of this talk is to explain the combinatorics of (K-theoretic) lacing diagrams and carefully state the formula. Time permitting, I will give some idea of the Gröbner degeneration technique used in the proof.
This video was produced by Syracuse University Department of Mathematics as part of ICRA 2016.
