Allen Knutson
(UC Berkeley)
Positive formulae for quiver polynomials
Abstract: Fix an isomorphism type r of representation of the equioriented A_n quiver, and consider the variety X_r of representations with that isomorphism type (or limits thereof). Buch and Fulton defined the quiver polynomials to be the multidegrees of these quiver cycles, and gave two formulae for them, one proven but alternating, the other conjectural but positive.
We give four formulae (all positive) for these quiver polynomials, one of which establishes the Buch-Fulton conjecture, the others being more efficient. The main ingredients are degenerations, the Gröbner geometry of [Knutson-Miller], and new form of the Zelevinsky map, and the combinatorics of key polynomials.
This is joint work with Ezra Miller (MSRI) and Mark Shimozono (Virginia Tech).