Ideals generated by superstandard tableaux

Abstract

We investigate products J of ideals of "row initial" minors in the polynomial ring K[X] defined by a generic m-by-n matrix. Such ideals are shown to be generated by a certain set of standard bitableaux that we call superstandard. These bitableaux form a Gr\"obner basis of J, and J has a linear minimal free resolution. These results are used to derive a new generating set for the Grothendieck group of finitely generated (T_m x GL_n(K))-equivariant modules over K[X]. We employ the Knuth--Robinson--Schensted correspondence and a toric deformation of the multi-Rees algebra that parameterizes the ideals J.