A SAGBI basis for the Quantum Grassmannian

Frank Sottile

   The maximal minors of a generic rectangular matrix satisfy an interesting
set of quadratic P\"ucker relations.  These are elegantly described in terms
of a natural Bruhat order defined on the set of maximal minors.  The
combinatorics of these relations and the Bruhat order has important
consequences for the algebra generated by the maximal minors, and for the
geometry of Grassmann varieties.

If we now consider a matrix of polynomials in a variable t, then the maximal
minors are themselves polynomials in t.  The goal of this talk is to
describe quadratic relations among the coefficients of t in these maximal
minors, and some of the consequences of the relations we find.  We will
begin by describing the classical Pl\"ucker relations and their
consequences.  This talk represents joint work with Bernd Sturmfels.