Ideals generated by superstandard tableaux

TL;DR

This paper studies ideals generated by superstandard tableaux in polynomial rings, showing they form a Gröbner basis and have linear resolutions, with applications to equivariant module classification.

Contribution

It introduces superstandard tableaux as generators for certain ideals, proving they form a Gröbner basis and yield linear resolutions, advancing algebraic and combinatorial understanding.

Findings

Superstandard tableaux generate the ideals of row initial minors.

These generators form a Gröbner basis for the ideals.

The ideals have linear minimal free resolutions.

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.