On the Reduced Gröbner Bases of Blockwise Determinantal Ideals

Abstract

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 Gröbner 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 Gröbner bases when the defining permutations are vexillary, in the non-vexillary case we derive an explicit formula for computing the reduced Gröbner 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 Gröbner bases, of Schubert determinantal ideals are also proven.

Figures (8)

• Figure 1: An illustrative example of the Rothe diagram and essential set of the permutation $w=[10, 9, 2, 3, 8, 6, 5, 7, 4, 1]$
• Figure 2: An illustrative example of a Fulton generator attending a submatrix
• Figure 3: Two-sided and one-sided ladders in a generic square matrix of size 9
• Figure 4: Illustration of $\mathop{\mathrm{lt}}\nolimits(G)$ and $\mathop{\mathrm{lt}}\nolimits(G_{\ell})$ in the proof of Proposition \ref{['prop:rowcolumn']} (left); Illustration of the two minors in $I_w$ with $w=2143$ in Example \ref{['example:notGB']} (right)
• Figure 5: Illustration of the block structures in Corollary \ref{['cor:uniqueuppercorner']}

Theorems & Definitions (75)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Theorem 2.6: KM05gKMY09g
• Definition 2.7
• Example 2.8
• Proposition 2.9: GY22M
• Theorem 2.10: GY22M