Blockwise determinantal ideals are those generated by the union of all the minors of specified sizes in certain blocks of a generic matrix, and they are the natural generalization of many existing determinantal ideals like the Schubert and ladder ones. In this paper we establish several criteria to verify whether the Gr\"obner bases of blockwise determinantal ideals with respect to (anti-)diagonal term orders are minimal or reduced. In particular, for Schubert determinantal ideals, while all the elusive minors form the reduced Gr\"obner bases when the defining permutations are vexillary, in the non-vexillary case we derive an explicit formula for computing the reduced Gr\"obner basis from elusive minors which avoids all algebraic operations. The fundamental properties of being normal and strong for W-characteristic sets and characteristic pairs, which are heavily connected to the reduced Gr\"obner bases, of Schubert determinantal ideals are also proven.

翻译：分块行列式理想是由一般矩阵特定块中指定大小的所有子式之并集生成的理想，是许多现有行列式理想（如Schubert行列式理想和梯子行列式理想）的自然推广。本文建立了若干准则，用于验证关于（反）对角项序的分块行列式理想的Gröbner基是否为极小或约化的。特别地，对于Schubert行列式理想而言，当定义置换为Vexillary型时，所有难解子式构成约化Gröbner基；而在非Vexillary情形下，我们从难解子式出发推导出计算约化Gröbner基的显式公式，该过程完全避免了代数运算。此外，本文还证明了与Schubert行列式理想约化Gröbner基密切相关的W-特征集与特征对的规范性及强性质。