Combinatorics of quiver polynomialsAlgebra & Discrete Mathematics
|Ezra Miller, Univ of Minnesota and MSRI Berkeley
|Thu, Apr 24 2003, 12:00PM
Buch and Fulton defined quiver polynomials in a geometric context, and proved that these polynomials can be expressed as integer sums of products of Schur functions in differences of alphabets. Motivated by the fact that Littlewood-Richardson numbers are (very) special cases of the integer `quiver coefficients' appearing in these sums, Buch and Fulton conjectured that all quiver coefficients are positive, and moreover described certain `factor sequenes' of Young tableaux that they should count. After indicating in an elementary way how quiver polynomials are defined via geometry, I will discuss combinatorial formulae for them that arise on the way to a solution of the Buch-Fulton conjecture. I will define everything from scratch, although previous familiarity with Schur functions will help in understanding the combinatorial motivations. This talk oncerns joint work with Allen Knutson and Mark Shimozono.