Ideals generated by superstandard tableaux

Summary

We investigate products J of ideals of “row initial” minors in the polynomial ring K[X] defined by a generic m × n-matrix. The defining “shape” of J determines a set of “row initial” standard bitableaux that we call superstandard. They form a Gröbner 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 Tm ×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 parametrizes the ideals J.

Let K be a field and X an m ×n matrix of indeterminates xi j over K. We write R = K[X] for the polynomial ring in the xi j . The group GL_m(K)×GL_n(K) acts on R with an action induced by the rule (g, g) · X = gXg⁻¹. The representation theory of R as a module for this group is intimately connected to the linear basis of R given by bitableaux [Bruns and Vetter 1988, Chapter 11; de Concini et al. 1980]. The bitableaux are products of minors which are indexed by pairs of tableaux of the same shape with strictly increasing rows and weakly increasing columns. We say that a bitableau is superstandard if its left factor tableau has column i filled with the number i. The left tableau determines the row indices of the minors whose product the bitableau represents.

For each i, 1≤i ≤m, let J_i ⊂ R denote the ideal generated by the size i minors of the first i rows of X. In the current work we study an arbitrary product of such ideals. For a decreasing sequence of positive integers min(m, n) ≥ s₁ ≥· · ·≥s_ν we set J_S = J_s1 . . . J_sν . It is a consequence of Theorem 2.2 that the ideals JS are exactly those that are generated by superstandard bitableau of shape S.