The University of Michigan Combinatorics Seminar
Winter 2001
March 23, 4:10-5:00, 3866 East Hall

A SAGBI basis for the quantum Grassmannian

Frank Sottile

University of Massachusetts-Amherst


Abstract

The maximal minors of a generic rectangular matrix satisfy interesting quadratic relations. A reduced Gröbner basis for the Plücker ideal of these relations is elegantly described in terms of a natural Bruhat order defined on the set of maximal minors. This shows the maximal minors generate an algebra with straightening law on this Bruhat order, and these results are classical.

   If we now consider a generic matrix of polynomials in a variable t, then the maximal minors are themselves polynomials in t. The goal of this talk is to describe a reduced Gröbner basis for the ideal of algebraic relations among these coefficients. This will show that the coordinate ring of the quantum Grassmannian (a singular compactification of the space of rational curves in the Grassmannian) is an algebra with straightening law on a quantum Bruhat order.

   We will begin by reviewing the classical situation described above, including some interesting geometric consequences of the reduced Gröbner basis for the Plücker ideal. We will then describe a reduced Gröbner basis for the quantum Plücker ideal, indicate our method of proof, and describe some consequences of these relations. This talk represents joint work with Bernd Sturmfels.

The quantum Bruhat order for degree 1 curves in the Grassmannian of 2-planes in 5-space.