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Minimal primes and radicality of ideals generated by adjacent 2-minors

Takayuki Hibi, Francesco Navarra, Ayesha Asloob Qureshi, Sara Saeedi Madani

TL;DR

The paper provides a complete description of the minimal primes of ideals generated by adjacent $2$-minors in terms of admissible sets and lattice ideals, and shows that unmixedness, Cohen–Macaulayness, level, Gorensteinness, and complete intersection are equivalent for these ideals. It then gives a precise combinatorial characterization in the convex case: a collection of cells yields any of these equivalent properties exactly when it contains no square tetromino or $X$-pentomino. The radicality of adjacent $2$-minor ideals is analyzed, with a formula for the radical as an intersection over admissible-set primes and the construction of infinite families of minimally non-radical examples, illustrating the nuanced interplay between combinatorics of cell configurations and algebraic properties. Overall, the work connects polyomino combinatorics to deep structural questions in determinantal-type ideals, offering both theoretical characterizations and computational insights into radicality and primes.

Abstract

In this paper, we provide a complete description of the minimal primes of ideals generated by adjacent $2$-minors, in terms of the so-called admissible sets and associated lattice ideals. We prove that for these ideals, the properties of being unmixed, Cohen-Macaulay, level, Gorenstein, and complete intersection are equivalent. Moreover, we give a combinatorial characterization of all convex collections of cells satisfying any of these equivalent properties. Finally, we study the radicality of these ideals and derive necessary combinatorial conditions based on minimal non-radical configurations.

Minimal primes and radicality of ideals generated by adjacent 2-minors

TL;DR

The paper provides a complete description of the minimal primes of ideals generated by adjacent -minors in terms of admissible sets and lattice ideals, and shows that unmixedness, Cohen–Macaulayness, level, Gorensteinness, and complete intersection are equivalent for these ideals. It then gives a precise combinatorial characterization in the convex case: a collection of cells yields any of these equivalent properties exactly when it contains no square tetromino or -pentomino. The radicality of adjacent -minor ideals is analyzed, with a formula for the radical as an intersection over admissible-set primes and the construction of infinite families of minimally non-radical examples, illustrating the nuanced interplay between combinatorics of cell configurations and algebraic properties. Overall, the work connects polyomino combinatorics to deep structural questions in determinantal-type ideals, offering both theoretical characterizations and computational insights into radicality and primes.

Abstract

In this paper, we provide a complete description of the minimal primes of ideals generated by adjacent -minors, in terms of the so-called admissible sets and associated lattice ideals. We prove that for these ideals, the properties of being unmixed, Cohen-Macaulay, level, Gorenstein, and complete intersection are equivalent. Moreover, we give a combinatorial characterization of all convex collections of cells satisfying any of these equivalent properties. Finally, we study the radicality of these ideals and derive necessary combinatorial conditions based on minimal non-radical configurations.
Paper Structure (4 sections, 14 theorems, 30 equations, 13 figures, 2 tables)

This paper contains 4 sections, 14 theorems, 30 equations, 13 figures, 2 tables.

Table of Contents

  1. Preliminaries
  2. Minimal primes of adjacent $2$-minor ideals
  3. From Unmixedness to Complete Intersection for adjacent $2$-minor ideals
  4. On the radicality of adjacent $2$-minor ideals

Key Result

Theorem 1

Let $\mathcal{C}$ be a collection of cells, and let $I_{\mathrm{adj}}(\mathcal{C})\subset S_{\mathcal{C}}=K[x_v \mid v \text{ is a vertex of } \mathcal{C}]$ denote its adjacent $2$--minor ideal. Then the following conditions are equivalent: Moreover, if $I_{\mathrm{adj}}(\mathcal{C})$ satisfies above conditions, then $\mathrm{ht}(I_{\mathrm{adj}}(\mathcal{C})) =|\mathcal{C}|$.

Figures (13)

  • Figure 1: A weakly connected collection of cells and a polyomino.
  • Figure 2: A square tetromino and an $X$-pentomino
  • Figure 3: Notation for the proof of Lemma \ref{['Coro: necessary condition for CI']}
  • Figure 4: Notation of the $X$-pentomino used in the proof of Lemma \ref{['Coro: necessary condition for CI with X pento']}.
  • Figure 5: A convex polyomino and a parallelogram.
  • ...and 8 more figures

Theorems & Definitions (36)

  • Theorem 1: Theorem \ref{['Thm: Unmixed = CI']}
  • Theorem 2: Theorem \ref{['Thm: characterization convex CI']}
  • Example 1.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • ...and 26 more