Matrix Schubert Varieties

• Matrix Schubert varieties are affine degeneracy loci defined by northwest rank conditions and Rothe diagrams, capturing permutation combinatorics through essential sets.
• Their determinantal ideals and minimal generators form Gröbner bases under antidiagonal and diagonal term orders, revealing deep combinatorial and geometric structures.
• They bridge algebraic geometry, combinatorics, and representation theory, with applications to Schubert polynomials, Gaussian graphical models, and toric degenerations.

Matrix Schubert varieties are affine degeneracy loci attached to permutations and partial permutations. For $w\in S_n$, they can be defined either as the Zariski closure of the $(B_-\times B_+)$-orbit of the permutation matrix $M_w$ in $M_n$, or as the locus of matrices whose northwest submatrices satisfy the rank bounds

$$r_w(i,j)=\#\{\,k\le i:\, w(k)\le j\,\},\qquad X_w=\left\{M\in M_{n\times n}:\operatorname{rank}(M_{[1..i],[1..j]})\le r_w(i,j)\ \text{for all }i,j\right\}.$$

Their defining ideals are Schubert determinantal ideals generated by minors, and they provide the affine model underlying much of the Gröbner geometry, equivariant geometry, and combinatorics of Schubert polynomials (Stelzer, 2023, Hamaker et al., 2020).

1. Foundational construction

Let $Z=(z_{ij})$ be the generic $n\times n$ matrix and $R=k[Z]$. The Schubert determinantal ideal associated to $w\in S_n$ is

$$I_w=\left\langle\text{all minors of size }r_w(i,j)+1\text{ of }Z_{[1..i],[1..j]}\text{ for all }i,j\right\rangle.$$

Equivalently, $(B_-\times B_+)$0 is the affine scheme cut out by the northwest rank conditions. In orbit-theoretic language, if $(B_-\times B_+)$1 and $(B_-\times B_+)$2 are the lower- and upper-triangular groups, then $(B_-\times B_+)$3 (Stelzer, 2023).

The combinatorics is organized by the Rothe diagram

$(B_-\times B_+)$4

and by the essential set $(B_-\times B_+)$5, consisting of southeast corners of connected components of $(B_-\times B_+)$6. Fulton showed that one may generate $(B_-\times B_+)$7 using only essential positions: $(B_-\times B_+)$8 He also showed that $(B_-\times B_+)$9 is prime, so $M_w$0 is reduced and irreducible (Hamaker et al., 2020).

For permutations, the codimension is $M_w$1, the Coxeter length, and hence

$M_w$2

The same rank-inequality formalism extends to partial permutations in rectangular matrix spaces; after extending a partial permutation $M_w$3 to a genuine permutation $M_w$4, one has $M_w$5. This reduction places many structural questions for partial permutations inside the square case (Hsiao, 2013).

2. Determinantal ideals and minimal equations

Fulton’s essential generators are generally not minimal. Gao and Yong introduced a finer selection rule based on three notions for an essential minor $M_w$6: it belongs to an essential position $M_w$7 if $M_w$8, $M_w$9, and $M_n$0; it attends a lower-rank essential position if its row and column sets force vanishing by cofactor expansion against smaller essential conditions; and it is elusive if it does not attend any lower-rank essential position. Their main theorem states that $M_n$1 is minimally generated by the elusive minors (Gao et al., 2022).

This description produces canonical generators indexed by boxes of the Rothe diagram. If $M_n$2 and $M_n$3, then the “corner minor”

$M_n$4

is elusive. In particular, there is always at least one minimal generator attached to each Rothe-diagram box. In the Grassmannian case, where $M_n$5 has a single element, every essential minor is automatically elusive, so Fulton’s list is already minimal (Gao et al., 2022).

Under any antidiagonal term order, Gao–Yong’s minimal generating set remains a Gröbner basis. Stelzer showed that if a degree-$M_n$6 element occurs in the reduced Gröbner basis under such a term order, then it has at least $M_n$7 terms; this lower bound is sharp. The same paper proved that $M_n$8 is binomial if and only if $M_n$9 avoids the patterns $$r_w(i,j)=\#\{\,k\le i:\, w(k)\le j\,\},\qquad X_w=\left\{M\in M_{n\times n}:\operatorname{rank}(M_{[1..i],[1..j]})\le r_w(i,j)\ \text{for all }i,j\right\}.$$0 and $$r_w(i,j)=\#\{\,k\le i:\, w(k)\le j\,\},\qquad X_w=\left\{M\in M_{n\times n}:\operatorname{rank}(M_{[1..i],[1..j]})\le r_w(i,j)\ \text{for all }i,j\right\}.$$1, and that Gao–Yong’s minimal Gröbner basis is reduced if and only if $$r_w(i,j)=\#\{\,k\le i:\, w(k)\le j\,\},\qquad X_w=\left\{M\in M_{n\times n}:\operatorname{rank}(M_{[1..i],[1..j]})\le r_w(i,j)\ \text{for all }i,j\right\}.$$2 is vexillary, i.e. $$r_w(i,j)=\#\{\,k\le i:\, w(k)\le j\,\},\qquad X_w=\left\{M\in M_{n\times n}:\operatorname{rank}(M_{[1..i],[1..j]})\le r_w(i,j)\ \text{for all }i,j\right\}.$$3-avoiding (Stelzer, 2023).

3. Antidiagonal Gröbner geometry and Schubert polynomials

The foundational Gröbner picture uses antidiagonal term orders: for every minor, the initial term is the antidiagonal monomial. Knutson and Miller proved that under any such order the Fulton generators form a Gröbner basis for $$r_w(i,j)=\#\{\,k\le i:\, w(k)\le j\,\},\qquad X_w=\left\{M\in M_{n\times n}:\operatorname{rank}(M_{[1..i],[1..j]})\le r_w(i,j)\ \text{for all }i,j\right\}.$$4. Consequently, $$r_w(i,j)=\#\{\,k\le i:\, w(k)\le j\,\},\qquad X_w=\left\{M\in M_{n\times n}:\operatorname{rank}(M_{[1..i],[1..j]})\le r_w(i,j)\ \text{for all }i,j\right\}.$$5 is squarefree, and the initial scheme is a union of coordinate subspaces (Hamaker et al., 2020).

These irreducible components are indexed by ordinary pipe dreams, or RC-graphs, for $$r_w(i,j)=\#\{\,k\le i:\, w(k)\le j\,\},\qquad X_w=\left\{M\in M_{n\times n}:\operatorname{rank}(M_{[1..i],[1..j]})\le r_w(i,j)\ \text{for all }i,j\right\}.$$6, each with multiplicity $$r_w(i,j)=\#\{\,k\le i:\, w(k)\le j\,\},\qquad X_w=\left\{M\in M_{n\times n}:\operatorname{rank}(M_{[1..i],[1..j]})\le r_w(i,j)\ \text{for all }i,j\right\}.$$7. This converts the geometry of matrix Schubert varieties into the pipe-dream formula for double Schubert polynomials: $$r_w(i,j)=\#\{\,k\le i:\, w(k)\le j\,\},\qquad X_w=\left\{M\in M_{n\times n}:\operatorname{rank}(M_{[1..i],[1..j]})\le r_w(i,j)\ \text{for all }i,j\right\}.$$8 Setting $$r_w(i,j)=\#\{\,k\le i:\, w(k)\le j\,\},\qquad X_w=\left\{M\in M_{n\times n}:\operatorname{rank}(M_{[1..i],[1..j]})\le r_w(i,j)\ \text{for all }i,j\right\}.$$9 gives the single Schubert polynomial. The identification proceeds through $Z=(z_{ij})$0-equivariant classes: the class of $Z=(z_{ij})$1 equals the class of its antidiagonal initial scheme, and additivity over coordinate components recovers the combinatorial expansion (Hamaker et al., 2020).

This antidiagonal geometry also supports alternative positive models. The prism tableau model expresses $Z=(z_{ij})$2 as a weight generating function over minimal prism tableaux without unstable triples, and its proof proceeds by comparing antidiagonal Gröbner degenerations, plus diagrams, and overlays of biGrassmannian pieces. In the Grassmannian case, the construction reduces to semistandard Young tableaux and recovers the Schur polynomial (Weigandt et al., 2015).

4. Diagonal degenerations, bumpless pipe dreams, and fine multidegrees

Diagonal term orders are defined dually: for every minor, the initial term is the main diagonal monomial. In contrast with the antidiagonal case, Fulton generators are not a Gröbner basis in general under diagonal orders, and the initial ideals can be nonreduced and depend on the specific diagonal order (Hamaker et al., 2020).

To repair this, Hamaker, Pechenik, and Weigandt introduced the CDG generators. Writing

$Z=(z_{ij})$3

one first sets $Z=(z_{ij})$4 on $Z=(z_{ij})$5, and then takes minors only in the specialized northwest submatrices attached to $Z=(z_{ij})$6. For banner permutations, these CDG generators form a diagonal Gröbner basis, and the diagonal initial ideal becomes

$Z=(z_{ij})$7

where $Z=(z_{ij})$8 denotes the set of bumpless pipe dreams for $Z=(z_{ij})$9. The same description holds for vexillary and predominant families, linking diagonal degenerations to Lam–Lee–Shimozono’s bumpless formula and to Lascoux’s $n\times n$0-vertex ice model (Hamaker et al., 2020).

Klein proved the full pattern-avoidance characterization conjectured in that work: the CDG generators form a diagonal Gröbner basis under every diagonal term order if and only if $n\times n$1 avoids eight explicit patterns, namely

$n\times n$2

This identifies the precise CDG class and extends the bumpless-pipe-dream description from banner permutations to all permutations in that avoidance class (Klein, 2020).

A further refinement is provided by fine multidegrees. Embedding $n\times n$3 into $n\times n$4 and taking the closure $n\times n$5, one obtains a squarefree homogeneous polynomial $n\times n$6, the fine Schubert polynomial, recording the multidegree of $n\times n$7. A general criterion then yields universal Gröbner bases from fine multidegrees: for permutations whose Schubert polynomials and inverse Schubert polynomials both have only $n\times n$8–$n\times n$9 coefficients, an explicitly enlarged set $R=k[Z]$0, built from CDG generators and merge relations, is a universal Gröbner basis for $R=k[Z]$1, and every initial ideal is reduced and squarefree (Huang et al., 2024).

5. Torus actions, toric factors, and complexity

Matrix Schubert varieties admit a canonical splitting into a nontrivial torus-geometric factor and a free affine factor. In the northwest conventions of Escobar and Mészáros, one writes

$R=k[Z]$2

where $R=k[Z]$3 is obtained from the northwest region by deleting the Rothe diagram. The factor $R=k[Z]$4 is toric with respect to the effective $R=k[Z]$5-action if and only if $R=k[Z]$6 is a union of disjoint hooks that do not share a row or a column (Escobar et al., 2015).

When $R=k[Z]$7 is toric, its moment polytope is a type-$R=k[Z]$8 root polytope associated to a bipartite graph $R=k[Z]$9, and its noncrossing alternating triangulation gives a geometric realization of a family of subword complexes. Portakal further identified toric $w\in S_n$0 with edge-ideal toric varieties of bipartite graphs and proved that $w\in S_n$1 is rigid if and only if every $w\in S_n$2-dimensional face of the associated cone $w\in S_n$3 is simplicial; this criterion can be reformulated directly in terms of the essential boxes of the Rothe diagram (Portakal, 2020).

For the usual diagonal torus action, the complexity of the essential factor is

$w\in S_n$4

where $w\in S_n$5 is the weight cone generated by the weights $w\in S_n$6 on the surviving coordinates. Recent work determines the full range of possibilities: for fixed $w\in S_n$7, the complexity of $w\in S_n$8 can be any integer in

$w\in S_n$9

and complexity $$I_w=\left\langle\text{all minors of size }r_w(i,j)+1\text{ of }Z_{[1..i],[1..j]}\text{ for all }i,j\right\rangle.$$0 does not occur. The maximum $$I_w=\left\langle\text{all minors of size }r_w(i,j)+1\text{ of }Z_{[1..i],[1..j]}\text{ for all }i,j\right\rangle.$$1 is achieved uniquely by $$I_w=\left\langle\text{all minors of size }r_w(i,j)+1\text{ of }Z_{[1..i],[1..j]}\text{ for all }i,j\right\rangle.$$2 (Escobar et al., 30 Sep 2025).

The same graph-cone formalism has been extended to Fulton’s opposite conventions and to Kazhdan–Lusztig varieties. In that setting one again isolates a factor $$I_w=\left\langle\text{all minors of size }r_w(i,j)+1\text{ of }Z_{[1..i],[1..j]}\text{ for all }i,j\right\rangle.$$3, studies how the weight cone changes under multiplication by simple reflections, and tracks complexity along Bruhat chains for Kazhdan–Lusztig varieties via cyclomatic numbers of associated graphs (Neuhaus et al., 30 Sep 2025).

6. Homological, singularity, and conormal properties

Matrix Schubert varieties are normal and Cohen–Macaulay, and their local geometry is closely tied to that of ordinary Schubert varieties. Hsiao gave a direct proof that matrix Schubert varieties are $$I_w=\left\langle\text{all minors of size }r_w(i,j)+1\text{ of }Z_{[1..i],[1..j]}\text{ for all }i,j\right\rangle.$$4-rational, avoiding Bott–Samelson resolutions. In characteristic $$I_w=\left\langle\text{all minors of size }r_w(i,j)+1\text{ of }Z_{[1..i],[1..j]}\text{ for all }i,j\right\rangle.$$5, this yields rational singularities. The same paper also characterized complete intersections: $$I_w=\left\langle\text{all minors of size }r_w(i,j)+1\text{ of }Z_{[1..i],[1..j]}\text{ for all }i,j\right\rangle.$$6 is a complete intersection if and only if, for every positive-rank box $$I_w=\left\langle\text{all minors of size }r_w(i,j)+1\text{ of }Z_{[1..i],[1..j]}\text{ for all }i,j\right\rangle.$$7, the associated northwest block $$I_w=\left\langle\text{all minors of size }r_w(i,j)+1\text{ of }Z_{[1..i],[1..j]}\text{ for all }i,j\right\rangle.$$8 is a permutation matrix and the smaller matrix Schubert variety $$I_w=\left\langle\text{all minors of size }r_w(i,j)+1\text{ of }Z_{[1..i],[1..j]}\text{ for all }i,j\right\rangle.$$9 is itself a complete intersection (Hsiao, 2013).

Castelnuovo–Mumford regularity is governed by Grothendieck polynomials. Rajchgot, Robichaux, and Weigandt proved

$(B_-\times B_+)$00

where $(B_-\times B_+)$01, the Rajchgot index, is the degree of the highest-degree homogeneous part of the Grothendieck polynomial $(B_-\times B_+)$02. They further identified the leading term of the corresponding Castelnuovo–Mumford polynomial with the Rajchgot code of $(B_-\times B_+)$03 and established weak-order and major-index characterizations of $(B_-\times B_+)$04 (Pechenik et al., 2021).

For covexillary permutations, Singh constructed an open embedding of a covexillary matrix Schubert variety into a Grassmannian Schubert variety. This yields an algebraic criterion for the conormal variety of a covexillary matrix Schubert variety in terms of rank conditions on an explicit block matrix built from a point $(B_-\times B_+)$05 and a cotangent vector $(B_-\times B_+)$06, and it implies that characteristic cycles of covexillary Schubert varieties are irreducible. The same embedding gives a new proof of Lascoux’s Grassmannian model for Kazhdan–Lusztig polynomials in the covexillary setting (Singh, 2022).

7. Variants, representation theory, and related models

Matrix Schubert varieties also support richer group actions than the diagonal torus. Under suitable Levi subgroups, their coordinate rings become bicrystalline representations. Using antidiagonal standard monomials and a filtered version of the Robinson–Schensted–Knuth correspondence, one obtains an explicit multiplicity formula for the irreducible decomposition of $(B_-\times B_+)$07; in the torus specialization this recovers the multigraded Hilbert-series formula of Knutson and Miller (Price et al., 2024).

Several structured variants have been developed. In the symmetric and upper-triangular settings, one obtains Schubert-type determinantal ideals compatible with diagonal Gröbner bases, and these models connect directly to Gaussian conditional independence and graphical models. In particular, certain “north-east” Gaussian conditional independence ideals and generalized Markov-chain models are realized as symmetric matrix Schubert varieties, which leads to combinatorial primary decomposition procedures via Bruhat order (Fink et al., 2015).

There is also a skew-symmetric analogue. Skew-symmetric matrix Schubert varieties are defined as nonempty intersections of ordinary matrix Schubert varieties with the subspace of skew-symmetric matrices. Their defining ideals are generated by Pfaffians rather than minors, and under graded reverse lexicographic order they admit Gröbner bases whose squarefree initial ideals decompose over fixed-point-free involution pipe dreams. This gives a geometric proof of explicit formulas for symplectic Grothendieck polynomials (Marberg et al., 2020).

At the level of intersections, Weigandt’s ASM varieties are arbitrary intersections of matrix Schubert varieties indexed by alternating sign matrices. They retain determinantal presentations and radical antidiagonal initial ideals, but irreducibility, equidimensionality, and Cohen–Macaulayness become subtler. Recent work proves codimension additivity under direct sum and formulates a stabilization conjecture for Cohen–Macaulayness under the operation $(B_-\times B_+)$08 (Axelrod-Freed et al., 15 May 2025).

A final extension moves to real geometry. For real matrix Schubert varieties, vexillarity is a necessary condition for the open dense regular locus to be minimal as a submanifold. Among vexillary partial permutations, minimality is proved for Grassmannian-type Rothe diagrams with at most two connected components, recovering all determinantal varieties and producing additional minimal cones (Lee et al., 3 Dec 2025).