Determinantal Ideals Overview

• Determinantal ideals are defined as polynomial ideals generated by the t×t minors of a matrix, capturing essential rank conditions in various algebraic structures.
• They possess well-understood Gröbner bases with squarefree initial ideals, which facilitate computational techniques and reveal fundamental structural properties.
• These ideals play a pivotal role in algebraic geometry, combinatorics, and statistics, offering diverse applications from degeneracy loci to combinatorial decompositions.

A determinantal ideal is a polynomial ideal generated by the minors of a matrix of indeterminates or specialized structure. These ideals serve as a central class of objects in commutative algebra, algebraic geometry, combinatorics, algebraic statistics, and representation theory. Determinantal ideals encode the vanishing of rank conditions on matrices, defining determinantal varieties and a wealth of related geometric and algebraic structures. This article provides a comprehensive overview of determinantal ideals and their generalizations, organized around their definitions, Gröbner bases, homological and structural properties, combinatorial frameworks, and modern applications.

1. Fundamental Definitions and Variants

A determinantal ideal is defined by fixing a matrix $X = (x_{ij})$ with entries in a polynomial ring $K[x_{ij}]$ and considering the ideal generated by its $t \times t$ minors: $$I_t(X) = \left\langle \text{all } t \times t \text{ minors of } X \right\rangle$$ For $X$ generic (i.e., with all entries independent variables), $I_t(X)$ cuts out the locus of matrices of rank $< t$ in the corresponding affine space.

Several important variants exist:

• Maximal minors: $t = \min(m, n)$, generating the ideal of invertible $X$.
• Partial determinantal ideals: Generators are selected minors, e.g., only those supported on prescribed blocks, ladders, or structural patterns (Mou et al., 2023).
• Determinantal facet ideals: Given a pure $(m-1)$-dimensional simplicial complex $A$ on $[n]$, the ideal $J_A$ is generated by maximal $m$-minors indexed by the facets of $A$ (Ene et al., 2011).
• Hypergraph determinantal ideals: Generators are $m$-minors indexed by $m$-uniform hyperedges, generalizing edge ideals to higher dimension (Mohammadi et al., 2012).
• Schubert determinantal ideals: Ideals defined via Fulton’s pattern (indexed by essential sets in permutations) generate matrix Schubert varieties (Stelzer et al., 2023, Mou et al., 2023).
• Specialized structures: Ideals arising from subregions of symmetric or Hankel matrices (e.g., Ferrers or s-Hankel hypermatrices) and ideals generated by selected slice minors (Sammartano, 2012, Corso et al., 2015).
• Coordinate section, blockwise, ladder, and matching field ideals: Generalizations where minors are taken over certain blocks/intervals, possibly with holes or overlaps (Mou et al., 2023, Mohammadi, 26 Mar 2024).

2. Gröbner Bases and Radicality

A central theme in the theory of determinantal ideals is the identification of term orders and combinatorial conditions guaranteeing that a given generating set (usually, selected minors) is a Gröbner basis.

• Classical result: For generic $X$, the $t$-minors form a Gröbner basis with respect to a diagonal (term-over-position) order, and the initial ideal is squarefree—hence $I_t(X)$ is radical (Ene et al., 2011, Mohammadi et al., 2012).
• Facet ideals and hypergraphs: For a determinantal facet ideal $J_A$ associated to an $(m-1)$-dimensional pure simplicial complex $A$, the set of $m$-minors is a Gröbner basis if and only if $A$ is closed in a precise combinatorial sense: the initial terms of any two generators corresponding to different facets with shared entries in the same position are relatively prime [(Ene et al., 2011), Theorem 1.1].
• Hypergraph closure: In the hypergraph setting, closure requires that for any two maximal cliques $\Delta_i, \Delta_j$ (blocks of minors), the minors from different blocks have leading terms which are pairwise coprime. This ensures the union of their Gröbner bases forms a Gröbner basis for the sum ideal [(Mohammadi et al., 2012), Proposition 2.6].
• Blockwise and Schubert ideals: In generalized blockwise settings (with overlaps or ladder shapes), explicit combinatorial criteria—based on anti-diagonal or diagonal term orders, lengths of overlaps, and the attending relationship between blocks—determine Gröbner basis minimality and reduction. For Schubert determinantal ideals, all Fulton generators are a Gröbner basis; for vexillary permutations, the elusive (non-attended) minors form the reduced basis (Mou et al., 2023).
• Specialized matrices: For Ferrers and s-Hankel matrices, the collection of all $2 \times 2$ minors indexed by the allowed positions forms a Gröbner basis with respect to suitable term orders. This result extends to specialized Ferrers ideals and is key to establishing radicality, Cohen–Macaulayness, and Koszulness (Corso et al., 2015, Sammartano, 2012).
• Sums and Knutson ideals: The closure properties of Knutson ideals ensure that for Knutson ideals associated to a squarefree initial term, the union of Gröbner bases for component ideals remains a Gröbner basis for the sum (Seccia, 2021, Seccia, 2020).
• Degeneration and monomial ideals: By selecting appropriate weight vectors (matching fields), determinantal ideals can degenerate to monomial ideals whose generators are images of minors. These initial ideals have the linear quotient property and identical Betti numbers for all choices of block structure (Mohammadi, 26 Mar 2024).

3. Primality, Decomposition, and Combinatorial Structure

While classical determinantal ideals are often prime, for generalized or mixed settings (such as facet, blockwise, or mixed minor ideals) primeness is more subtle and depends on the combinatorics.

• Criterion for primeness: For determinantal facet ideals, if the underlying complex $A$ is closed and the clique decomposition of $A$ satisfies that the intersection of vertex sets of any $t$ distinct cliques has size at most $m-t$, then $J_A$ can be prime [(Ene et al., 2011), Theorem 2.2].
• Hypergraph decomposition and prime splitting: Many determinantal hypergraph ideals decompose as sums of smaller ideals with the property that every minimal prime of the total ideal is the sum of minimal primes for the summands (a “prime splitting”). This is established for block adjacent hypergraphs and related constructions [(Mohammadi et al., 2012), Theorem 4.5].
• Non-primality in generalized settings: For s-Hankel hypermatrices ($s \geq 2$), the ideal generated by all $2 \times 2$ slice minors is not prime, and its minimal primes are described combinatorially as being indexed by maximal $(s, t)$-switchable sets (Sammartano, 2012). For CI ideals with hidden variables, a set of prime or radical components are indexed by combinatorial equivalence classes, with explicit formulas for their enumeration and generating sets (Alexandr et al., 5 May 2025).
• Equivalence classes and combinatorial types: For mixed determinantal ideals arising in algebraic statistics and related to structural zeros, combinatorial equivalence classes are defined by the distribution of zeros in the indexing grid, and the number of such classes is computed via closed formulas (Alexandr et al., 5 May 2025).
• Codeterminantal graphs: Two graphs are codeterminantal if their associated univariate determinantal ideals coincide, generalizing cospectral/coinvariant relationships in algebraic graph theory (Abiad et al., 2019).

4. Homological and Algebraic Properties

Determinantal ideals and their quotient rings exhibit a rich homological profile, closely linked to their combinatorial origin and Gröbner structure.

• Cohen–Macaulayness and normality: Determinantal ideals under suitable hypotheses (genericity, closure, appropriate term order) yield Cohen–Macaulay, normal, and often Gorenstein quotient rings. These properties are established in settings ranging from classical $I_t(X)$ ideals to determinantal facet, Schubert, and Ferrers ideals (Ene et al., 2011, Corso et al., 2015, Stelzer et al., 2023).
• Minimal free resolutions: The Eagon–Northcott complex provides the minimal free resolution for classical determinantal ideals. For hypergraph and facet variants, the minimal free resolution is obtained as the tensor product of the resolutions of component clique ideals. The linear strand of a determinantal facet ideal is given by the generalized Eagon–Northcott complex associated to its clique complex (Herzog et al., 2015, Mohammadi et al., 2012).
• Betti numbers and linear resolutions: For determinantal facet ideals $J_C$ generated by maximal minors indexed by an $m$-clutter, the graded Betti numbers on the linear strand are given by the combinatorial face numbers of the associated clique complex:

$$\beta_{i,i+m}(J_C) = \binom{m+i-1}{m-1} f_{m+i-1}(\Delta(C))$$

The ideal $J_C$ has a linear resolution if and only if the clutter $C$ is complete (Herzog et al., 2015).

• Koszulness, Rees algebras, and multi-Rees algebras: Under suitable settings (especially for “close cuts” of Hankel matrices or Ferrers ideals), the Rees algebra and special fiber rings are defined by quadratic Gröbner bases and are Koszul and Cohen–Macaulay (Jafari, 2018, Corso et al., 2015).
• Monomial degenerations and cellular resolutions: The monomial initial ideals arising from matching field degenerations have the linear quotient property (for suitable generator orderings) and admit minimal cellular resolutions supported on regular CW complexes. The Betti numbers are computed via explicit combinatorial formulas (Mohammadi, 26 Mar 2024).

5. Symbolic Powers, Singularities, and F-Theory

The behavior of symbolic powers of determinantal ideals has deep connections to singularity theory and the paper of F-singularities.

• Symbolic powers and regularity: For determinantal ideals of generic, symmetric, and Hankel matrices, and for Pfaffians of skew-symmetric matrices, the limit $\lim_{n\to\infty} \text{reg}(I^{(n)})/n$ exists, and accordingly, the depth of $R/I^{(n)}$ stabilizes for large $n$. Explicit formulas for this stabilized depth are given in the generic and skew-symmetric cases (Montaño et al., 2020, Stefani et al., 2021).
• F-purity, F-thresholds, and test ideals: In positive characteristic, the generalized test ideals associated to determinantal ideals of maximal minors are completely characterized by the formula $T(c \cdot I) = I^{c + 1 - \operatorname{fpt}(I)}$ for $c \geq \operatorname{fpt}(I) - 1$, where $\operatorname{fpt}(I)$ is the F-pure threshold. The jumping numbers coincide with the set $\{ n \geq \operatorname{fpt}(I) \}$, and it is conjectured that analogous statements hold for minors of arbitrary size (Henriques, 2014).
• F-split and F-regular blowup algebras: The Rees and symbolic Rees algebras of determinantal ideals are F-split or strongly F-regular in prime characteristic under explicit conditions, leading to bounds for regularity and strong control over blowup algebras (Stefani et al., 2021). Symbolic F-purity is developed and is characterized by a Fedder-type criterion.
• Knutson ideals and squarefree initial terms: The class of Knutson ideals, built from a polynomial with squarefree initial term, includes determinantal ideals of generic and Hankel matrices. This ensures that their initial ideals are squarefree, thus radical, and that their algebraic and combinatorial properties are preserved across characteristics (Seccia, 2020, Seccia, 2021).

6. Applications in Algebraic Geometry, Combinatorics, and Statistics

Determinantal ideals serve as a crucial algebraic tool in several domains.

• Geometry of degeneracy loci and classical varieties: Determinantal ideals define loci with prescribed rank conditions (e.g., Grassmannian, Schubert, Segre, and rational normal varieties) (Stelzer et al., 2023, Ene et al., 2011).
• Combinatorics and graph theory: Binomial edge ideals, as $2$-minors indexed by edges, encode conditional independence structures in graphical statistical models (Ene et al., 2011, Alexandr et al., 5 May 2025). Codeterminantal graphs generalize concepts like cospectral graphs via their determinantal invariants (Abiad et al., 2019).
• Algebraic statistics: Mixed determinantal ideals model conditional independence in the presence of hidden variables and allow for a refined decomposition indexed by combinatorial equivalence classes of structural zeros (Alexandr et al., 5 May 2025).
• Resolutions and invariant theory: The syzygies of determinantal ideals are acted upon by the general linear (super)group and have deep connections to the cohomology of flag supervarieties (Sam et al., 2021).
• Degenerations and toric geometry: Gröbner degenerations of determinantal ideals connect algebraic invariants to combinatorics via matching field ideals and toric degenerations of Grassmannians, preserving Betti numbers across different weight vectors (Mohammadi, 26 Mar 2024).

7. Representative Formulas and Structured Data

Several key formulas encapsulate the essential properties and invariants of determinantal ideals.

Context	Formula or Result	Reference
Height formula (generic)	$\operatorname{ht}(I_t(X)) = (n-t+1)(m-t+1)$	(Seccia, 2021)
Betti numbers (facet ideals)	$$\beta_{i,i+m}(J_C) = \binom{m+i-1}{m-1} f_{m+i-1}(\Delta(C))$$	(Herzog et al., 2015)
Regularity limit	$$\displaystyle\lim_{n\to\infty} \frac{\operatorname{reg}(I^{(n)})}{n}$$ exists	(Montaño et al., 2020, Stefani et al., 2021)
F-pure threshold (max minors)	$\operatorname{fpt}(I) = n - m + 1$, $T(c \cdot I) = I^{c + 1 - \operatorname{fpt}(I)}$ for $c \geq \operatorname{fpt}(I) - 1$	(Henriques, 2014)
Codecomposability (hypergrp.)	$J_\Delta = J_{\Delta_1} + \cdots + J_{\Delta_r}$ under closedness and prime splitting conditions	(Mohammadi et al., 2012)
Initial monomial ideals	$M_a = \operatorname{in}_w(I_X)$ for matching field weight $w$ and determinantal ideal $I_X$	(Mohammadi, 26 Mar 2024)
Schubert Hilbertian property	$H_R(k) = P_R(k)$ for all $k \geq 0$ if $R = S/I_w$ for Schubert determinantal ideal $I_w$, with regularity bound via effective region $|X(w)|$	(Stelzer et al., 2023)
Slice minor generators (s-H)	$$I(s, t) = \langle f_{k, a, b} : d(a, b) = 2, k \in [t] \rangle$$, $$f_{k, a, b} = x_a x_b - x_{\mathrm{sw}(k, a, b)} x_{\mathrm{sw}(k, b, a)}$$	(Sammartano, 2012)
Radical components (CI)	$$\sqrt{I_C} = \bigcap_{S \subset [k\ell]} \sqrt{I_S}$$, with $I_S$ combinatorially determined from zero structure	(Alexandr et al., 5 May 2025)

These formulas highlight the interplay between algebraic generators, combinatorial structure, and geometric and homological invariants throughout the landscape of determinantal ideal theory.

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The theory of determinantal ideals thus demonstrates a rich synergy between algebra, combinatorics, geometry, and computational structure, unified by minor-generating conditions and their combinatorial frameworks. Advances in this field continue to inform broad areas of mathematics, from the paper of syzygies and singularities to applications in statistical modeling and beyond.