Li's Double Determinantal Varieties

• Li’s double determinantal varieties are algebraic sets defined by imposing simultaneous rank conditions on horizontal and vertical concatenations of generic matrices.
• Fieldsteel and Klein proved that the natural generators form a Gröbner basis, establishing the varieties as normal, irreducible, and Cohen–Macaulay.
• Special toric cases offer a combinatorial description via triple Segre embeddings and Hibi rings, linking algebraic geometry with combinatorial commutative algebra.

Searching arXiv for the core paper and directly related work on double determinantal varieties, toric cases, and Gröbner bases. Li’s double determinantal varieties are algebraic varieties defined by imposing simultaneous determinantal rank conditions on two canonical concatenations of a family of generic matrices. In the formulation studied by Fieldsteel and Klein, one fixes integers $r,m,n \ge 2$, a perfect base field $K$, and $r$ generic $m \times n$ matrices $X_1,\dots,X_r$; one then forms the horizontal concatenation $H=(X_1\cdots X_r)$ and the vertical concatenation $$V=\begin{bmatrix}X_1\ \vdots \ X_r\end{bmatrix}$$, and defines the double determinantal ideal

$J=I_s(H)+I_t(V)$

for integers $s \le m$ and $t \le n$. The associated affine or projective variety is the double determinantal variety attached to $K$0 (Fieldsteel et al., 2019). These varieties were introduced by Li as a special case of Nakajima quiver varieties, and the central conjecture attached to them—normality, irreducibility, Cohen–Macaulayness, and Gröbner basis behavior of the natural determinantal generators—was proved in full generality by Fieldsteel–Klein (Fieldsteel et al., 2019). Subsequent work placed the construction in the broader framework of bipartite determinantal ideals (Illian et al., 2023), isolated the toric $K$1 minor case (Blose et al., 2020, Biermann et al., 28 Jun 2025), and contrasted Li’s notion with an unrelated usage of “double determinantal” for square thickenings $K$2 of classical determinantal ideals (Huang, 2020).

1. Definition and ambient construction

The basic data consist of integers $K$3, a perfect field $K$4, and matrices

$K$5

whose entries are distinct indeterminates. The coordinate ring is the standard graded polynomial ring

$K$6

The two concatenations are the $K$7 matrix

$K$8

and the $K$9 matrix

$r$0

For fixed integers $r$1 and $r$2, the double determinantal ideal is

$r$3

generated by all $r$4 minors of $r$5 together with all $r$6 minors of $r$7. These minors are the “natural generators” of $r$8 (Fieldsteel et al., 2019).

Geometrically, the corresponding variety is the common zero locus of the rank conditions $r$9 and $m \times n$0. In the projective formulation, it sits in $m \times n$1 with homogeneous coordinates given by the variables $m \times n$2 (Blose et al., 2020). The construction generalizes ordinary determinantal varieties by coupling two different structured flattenings of the same family of matrices. A plausible implication is that the geometry is governed simultaneously by horizontal and vertical interaction between the blocks, rather than by a single global matrix.

A term order $m \times n$3 on $m \times n$4 is called diagonal if, for every square submatrix of $m \times n$5 or $m \times n$6, the initial term of its determinant is the product of the entries on its main diagonal (Fieldsteel et al., 2019). Such orders exist; one example given in the literature is the graded lexicographic order induced by reading the entries of $m \times n$7 in row-major order (Fieldsteel et al., 2019). In the broader bipartite framework, a lexicographic monomial order consistent with every block matrix $m \times n$8 is likewise diagonal, in the sense that the leading term of every determinant is the NW–SE diagonal product (Illian et al., 2023).

2. Relation to Nakajima quiver varieties and bipartite determinantal ideals

Li introduced double determinantal varieties as a special case of Nakajima quiver varieties (Fieldsteel et al., 2019). Conceptually, the relevant quiver has representation data consisting of $m \times n$9 parallel $X_1,\dots,X_r$0 linear maps, and the two concatenations encode the representation space together with two families of rank conditions: horizontal rank $X_1,\dots,X_r$1 on $X_1,\dots,X_r$2 and vertical rank $X_1,\dots,X_r$3 on $X_1,\dots,X_r$4 (Fieldsteel et al., 2019). The commutative-algebraic study by Fieldsteel–Klein focuses not on the full quiver-theoretic structure, but on the defining ideals, Gröbner bases, and homological properties (Fieldsteel et al., 2019).

This perspective was later subsumed into the theory of bipartite determinantal ideals. For a bipartite quiver $X_1,\dots,X_r$5, one associates to each vertex $X_1,\dots,X_r$6 a block matrix $X_1,\dots,X_r$7, and defines the ideal $X_1,\dots,X_r$8 by the $X_1,\dots,X_r$9-minors of all $H=(X_1\cdots X_r)$0. When $H=(X_1\cdots X_r)$1 has one source, one sink, and $H=(X_1\cdots X_r)$2 arrows between them, the two central block matrices are precisely the two flattenings

$H=(X_1\cdots X_r)$3

and the resulting ideal $H=(X_1\cdots X_r)$4 is the two-vertex specialization coinciding with Li’s double determinantal ideal (Illian et al., 2023).

This broader formulation clarifies two features. First, Li’s varieties are not isolated constructions but instances of quiver-theoretic determinantal loci (Illian et al., 2023). Second, the same algebraic mechanism applies to tensors: the matrices $H=(X_1\cdots X_r)$5 and $H=(X_1\cdots X_r)$6 are two flattenings of a $H=(X_1\cdots X_r)$7-tensor, so the defining minors encode bounded rank on two tensor flattenings (Illian et al., 2023). The literature explicitly connects this to tensor rank, subspace varieties, and algebraic statistics, including independence and conditional independence models (Illian et al., 2023).

A distinct terminological issue arises in representation-theoretic work on determinantal thickenings. In that literature, “double determinantal variety” may refer to the square thickening $H=(X_1\cdots X_r)$8 of the classical determinantal variety defined by a single generic matrix (Huang, 2020). That usage is different from Li’s concatenation-based construction. The papers make clear that these are separate notions, even though both involve a “double” determinantal label (Fieldsteel et al., 2019, Huang, 2020).

3. Li’s conjecture and its resolution

Li conjectured that double determinantal varieties are normal, irreducible, and Cohen–Macaulay, and that their defining ideals admit a Gröbner basis consisting of the natural generators with respect to any diagonal term order (Fieldsteel et al., 2019). Fieldsteel and Klein proved this conjecture in full generality.

Their main results can be summarized as follows. Theorem 4.1 states that the natural generators of a double determinantal ideal form a Gröbner basis under any diagonal term order, and that double determinantal varieties are reduced, arithmetically Cohen–Macaulay, and glicci (Fieldsteel et al., 2019). Corollary 4.7 states that double determinantal varieties are normal and irreducible (Fieldsteel et al., 2019).

The proof strategy combines Gröbner basis arguments with liaison theory. Gröbner basisness is obtained by an inductive linkage argument generalizing earlier methods for mixed ladder determinantal varieties (Fieldsteel et al., 2019). Reducedness and Cohen–Macaulayness are deduced from the square-free structure of the initial ideal, and the glicci property is obtained from basic double $H=(X_1\cdots X_r)$9-links (Fieldsteel et al., 2019). Normality is then deduced from Serre’s criterion: the Jacobian ideal has height at least $$V=\begin{bmatrix}X_1\ \vdots \ X_r\end{bmatrix}$$0 in $$V=\begin{bmatrix}X_1\ \vdots \ X_r\end{bmatrix}$$1, giving regularity in codimension $$V=\begin{bmatrix}X_1\ \vdots \ X_r\end{bmatrix}$$2, while Cohen–Macaulayness supplies the $$V=\begin{bmatrix}X_1\ \vdots \ X_r\end{bmatrix}$$3 condition; hence $$V=\begin{bmatrix}X_1\ \vdots \ X_r\end{bmatrix}$$4 is normal, and since a normal graded ring is a domain, the variety is irreducible (Fieldsteel et al., 2019).

The broader bipartite determinantal literature later supplied an alternative proof of Gröbner basisness. Instead of liaison theory, the proof in the bipartite setting uses Buchberger’s criterion together with a direct $$V=\begin{bmatrix}X_1\ \vdots \ X_r\end{bmatrix}$$5-polynomial analysis based on the Leibniz formula for determinants (Illian et al., 2023). For the two-vertex specialization, this recovers the Gröbner basis result for Li’s varieties and emphasizes that the natural determinantal generators reduce all $$V=\begin{bmatrix}X_1\ \vdots \ X_r\end{bmatrix}$$6-pairs under a consistent diagonal lexicographic order (Illian et al., 2023). This does not replace the Fieldsteel–Klein arguments for normality and Cohen–Macaulayness, but it provides a distinct computational route to the initial ideal structure.

4. Gröbner bases, initial ideals, and liaison-theoretic structure

A key feature of Li’s double determinantal ideals is that the natural generators behave optimally with respect to diagonal term orders. For every generating determinant $$V=\begin{bmatrix}X_1\ \vdots \ X_r\end{bmatrix}$$7, the initial term is the product of the main diagonal entries of $$V=\begin{bmatrix}X_1\ \vdots \ X_r\end{bmatrix}$$8, and hence the initial ideal $$V=\begin{bmatrix}X_1\ \vdots \ X_r\end{bmatrix}$$9 is generated by square-free monomials corresponding to those diagonals (Fieldsteel et al., 2019). This square-free structure underlies reducedness and Cohen–Macaulayness via Stanley–Reisner theory; Remark 4.3 explicitly interprets the relevant simplicial complexes as vertex decomposable through link and deletion (Fieldsteel et al., 2019).

The proof proceeds first through a special case with two matrices and maximal minors. If $J=I_s(H)+I_t(V)$0 and $J=I_s(H)+I_t(V)$1 are $J=I_s(H)+I_t(V)$2 matrices, $J=I_s(H)+I_t(V)$3, $J=I_s(H)+I_t(V)$4, and $J=I_s(H)+I_t(V)$5, then the natural generators form a Gröbner basis under any diagonal term order; moreover, $J=I_s(H)+I_t(V)$6 is reduced and Cohen–Macaulay (Fieldsteel et al., 2019). The proof constructs auxiliary ideals $J=I_s(H)+I_t(V)$7 and $J=I_s(H)+I_t(V)$8, identifies their initial ideals using a lemma of Gorla–Migliore–Nagel, and concludes that the monomial ideal $J=I_s(H)+I_t(V)$9 generated by main diagonals equals $s \le m$0 (Fieldsteel et al., 2019).

The general case is organized through one-sided mixed ladders on $s \le m$1 and $s \le m$2, encoded by Young diagrams superimposed northwest on the concatenations. For ideals $s \le m$3 defined by mixed minors on these ladders, Claim 4.2 asserts that the natural generators form a Gröbner basis under any diagonal term order (Fieldsteel et al., 2019). The double determinantal case arises by specializing to full rectangular ladders.

The inductive step removes a carefully chosen boundary variable, constructs refined ladder ideals $s \le m$4, and produces module isomorphisms

$s \le m$5

using cofactor expansions that append the removed variable at southeast corners (Fieldsteel et al., 2019). Lemma 3.1 of Gorla–Migliore–Nagel then implies $s \le m$6 (Fieldsteel et al., 2019).

The liaison framework enters through basic double $s \le m$7-links. In the inductive construction one obtains $s \le m$8, where $s \le m$9 is a non-zerodivisor on $t \le n$0, so $t \le n$1 is a basic double $t \le n$2-link of $t \le n$3 on $t \le n$4 (Fieldsteel et al., 2019). Since the intermediate ideals are Cohen–Macaulay by induction, the new ideal remains Cohen–Macaulay; iterating the argument shows that double determinantal ideals are glicci, i.e. obtained from a complete intersection by finitely many direct algebraic $t \le n$5-links (Fieldsteel et al., 2019). This liaison-theoretic strengthening is one of the features distinguishing the Fieldsteel–Klein proof from later elementary $t \le n$6-pair proofs (Fieldsteel et al., 2019, Illian et al., 2023).

5. Dimension, height, and explicit formulas

Fieldsteel and Klein give an explicit height formula for the double determinantal ideal $t \le n$7 associated to $t \le n$8 with parameters $t \le n$9 and $K$00: $K$01 (Fieldsteel et al., 2019).

They also derive the projective dimension formula for the associated projective double determinantal variety $K$02: $K$03 (Fieldsteel et al., 2019).

The paper interprets these formulas inductively: the first term in the height is the height of $K$04, while the other terms count how often a new generator increases the height as southeast variables are removed in the linkage induction (Fieldsteel et al., 2019). Likewise, the projective dimension formula is described as the base dimension of the determinantal variety of $K$05-minors in one $K$06 matrix, plus contributions from variables that never occur as final entries in leading terms of newly introduced generators (Fieldsteel et al., 2019).

An explicit example appears when $K$07, $K$08, and $K$09. Writing the three matrices as $K$10, the projective variety has

$K$11

(Fieldsteel et al., 2019). The decomposition $K$12 is explained combinatorially in terms of entries of $K$13 and $K$14 that do not appear as terminal diagonal entries of relevant minors (Fieldsteel et al., 2019).

In the toric case $K$15, the formulas simplify drastically. The projective dimension becomes

$K$16

and the affine coordinate ring has Krull dimension $K$17 (Blose et al., 2020). The later Hibi-ring treatment recovers the same affine dimension formula

$K$18

by identifying the ring with the Hibi ring of a distributive lattice $K$19 whose underlying poset has cardinality $K$20 (Biermann et al., 28 Jun 2025).

6. Toric double determinantal varieties

The special case $K$21, equivalently $K$22, is toric. Here all generators are $K$23 minors, hence quadratic binomials, and the ideal is prime; therefore the associated double determinantal variety is toric (Blose et al., 2020). In this case one obtains the explicit parametrization

$K$24

with $K$25, $K$26, and $K$27, so the projective variety is the triple Segre embedding

$K$28

(Blose et al., 2020).

Theorem 3.1 of the toric study proves that $K$29 is prime, hence the projective variety is irreducible (Blose et al., 2020). Theorem 3.2 proves that every toric double determinantal variety is smooth (Blose et al., 2020). Since smooth projective toric varieties are automatically normal and Cohen–Macaulay, this recovers and strengthens, in the $K$30 regime, the general irreducibility and Cohen–Macaulay conclusions proved by Fieldsteel–Klein (Fieldsteel et al., 2019, Blose et al., 2020).

A later development identifies toric double determinantal rings with Hibi rings. Let $K$31 be the disjoint union of three chains of lengths $K$32, $K$33, and $K$34, and $K$35 its distributive lattice of order ideals. Then

$K$36

so toric double determinantal rings are Hibi rings (Biermann et al., 28 Jun 2025). This yields closed formulas for numerous invariants.

The explicit formulas established in that setting include:

Invariant	Formula	Source
$K$37	$K$38	(Biermann et al., 28 Jun 2025)
$K$39	$K$40	(Biermann et al., 28 Jun 2025)
$K$41	$K$42	(Biermann et al., 28 Jun 2025)
$K$43	$K$44	(Biermann et al., 28 Jun 2025)
$K$45	$K$46	(Biermann et al., 28 Jun 2025)

The same work gives the Hilbert function

$K$47

and the Hilbert series

$K$48

where $K$49 is expressed either through linear extensions of $K$50 or through the multiset Eulerian polynomial on $K$51 (Biermann et al., 28 Jun 2025).

The same paper answers a question of Li in the toric case by proving that $K$52 is Gorenstein if and only if each of $K$53 is either $K$54 or equals the common maximum; equivalently, all nonempty chains in $K$55 have equal length (Biermann et al., 28 Jun 2025).

7. Connections, variants, and related interpretations

Li’s double determinantal varieties sit at the intersection of determinantal geometry, quiver varieties, tensor methods, and combinatorial commutative algebra. In the quiver-theoretic direction, they provide explicit examples of bipartite determinantal loci arising from Nakajima’s graded quiver varieties (Illian et al., 2023). In the tensor direction, the two concatenations $K$56 and $K$57 are two flattenings of a $K$58-tensor, so the vanishing of their minors imposes tensor rank constraints. The bipartite determinantal paper makes this precise and extends the Gröbner-basis statement to unions of minors from two flattenings of higher-order tensors, as well as to some subspace varieties (Illian et al., 2023).

In algebraic statistics, the same flattening equations model independence and conditional independence conditions on probability tensors (Illian et al., 2023). The paper states that certain independence models correspond exactly to double determinantal ideals and discusses when two-flattening constraints render a third flattening redundant through the inequality

$K$59

(Illian et al., 2023). This suggests that Li’s construction is a natural algebraic interface between quiver rank conditions and statistical rank models.

There are also important contrasts. The representation-theoretic papers on determinantal thickenings use “double determinantal” for the nonreduced scheme $K$60, the square thickening of an ordinary determinantal variety (Huang, 2020). In that framework, one studies linear strands of minimal free resolutions via the BGG correspondence and $K$61-modules, decomposing $K$62 into principal $K$63-invariant ideals indexed by partitions

$K$64

(Huang, 2020). That body of work is algebraically rich, but it concerns a different geometric object from Li’s concatenation-defined varieties.

Another variant comes from jet schemes. For the classical rank-$K$65 determinantal variety, the principal component of the first jet scheme is described in separate literature as a “double determinantal variety,” characterized by vanishing of each $K$66 minor together with its first-order linearization (Ghorpade et al., 2012). The degree and Hilbert series of that principal component are the squares of those of the base Segre variety (Ghorpade et al., 2012). This is again a separate construction from Li’s, though the repeated terminology reflects a common determinantal doubling phenomenon.

Finally, the toric literature isolates a sharp dichotomy. In the concatenation framework, the double determinantal variety is toric precisely when $K$67 (Blose et al., 2020). If $K$68 and exactly one of $K$69 equals $K$70 while the other exceeds $K$71, then the ideal collapses to an ordinary determinantal ideal, so the only genuinely “double” toric case for $K$72 is $K$73 (Blose et al., 2020). This delineates the precise range in which toric and Hibi-ring methods apply directly.

Taken together, these results establish Li’s double determinantal varieties as a well-structured family of determinantal loci with unusually strong algebraic properties: their natural generators form Gröbner bases under diagonal orders; their coordinate rings are reduced, arithmetically Cohen–Macaulay, normal, and irreducible in full generality; and in the toric $K$74 case they admit a complete combinatorial description via triple Segre geometry and Hibi rings (Fieldsteel et al., 2019, Blose et al., 2020, Biermann et al., 28 Jun 2025).