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Complexity of the Zero Set of a Matrix Schubert Ideal

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Pith's one-line read For each n, the torus complexity of the essential slice of a matrix Schubert variety takes every integer value from 0 to (n−1)(n−3) except 1, and the maximum is achieved by exactly one permutation.

desk verdict The complexity classification is correct and new; minor expositional gaps don't threaten the result. read the letter →

arxiv 2510.00131 v3 pith:LCGGYDEJ submitted 2025-09-30 math.AG math.CO

Laura Escobar , Cesar Meza This is my paper
classification math.AGmath.CO MSC 14M1514L3014M12
keywords matrixSchubertvarietiesT-varietiestoruscomplexityoppositeRothediagramsedgeconespermutationsdeterminantalweightcone
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the torus complexity of the essential slice Y_w of a matrix Schubert variety — the part obtained after factoring out the largest affine space. Complexity measures how far a torus action is from being toric: complexity 0 means the variety is toric, and lower complexity means more combinatorial methods apply. The paper proves that for n ≥ 4 the maximum possible complexity is (n−1)(n−3), that the permutation [n, n−1, ..., 3, 1, 2] is the unique one reaching it, and that every integer from 0 to that maximum except 1 occurs as the complexity of some Y_w. Thus, for each fixed n, the set of achievable complexities is exactly {0, 2, 3, ..., (n−1)(n−3)}, answering a classification question that remained open after an earlier result ruled out complexity 1.

What carries the argument

The load-bearing identity is d_w = |L'(w)| − |V(G_w)| + |C(G_w)|. Here L'(w) is the region of boxes in the southwest diagram outside the opposite Rothe diagram, and G_w is the acyclic bipartite graph whose edges are the boxes of the L-diagram; |V| counts nonempty rows plus nonempty columns, |C| counts connected components. Complexity equals the number of free directions in Y_w minus the dimension of the weight cone, and the identity translates this into a purely combinatorial count. The maximum is then obtained by maximizing edges minus vertices over subgraphs of K_{n−1,n−1}, using the bound that each connected component contributes at most (r_i−1)(c_i−1), controlled by Cauchy–Schwarz.

What would settle it

Compute d_w = |L'(w)| − |V(G_w)| + |C(G_w)| for all 24 permutations in S_4. The theorem predicts the achievable set is {0, 2, 3} and the maximum 3 is attained only by [4, 3, 1, 2]; any other result for S_4, or any permutation in any S_n with complexity 1, would falsify the central claim.

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Extended reading notes

Core claim

Given a permutation w in S_n, the matrix Schubert variety X_w decomposes as Y_w × C^k with k maximal. The paper proves that with respect to the diagonal-torus action, the complexity d_w = dim(Y_w) − dim(torus orbit) equals |L'(w)| − |V(G_w)| + |C(G_w)|, where L'(w) is a certain diagram of boxes attached to the permutation and G_w is an acyclic bipartite graph built from it. The main theorem states that d_max(n) = (n−1)(n−3) for n ≥ 4, achieved uniquely by w0s_{n−1} = [n, n−1, n−2, ..., 3, 1, 2], and that every integer in {0, 2, 3, ..., (n−1)(n−3)} is realized. The proof bounds the complexity using the row and column counts of the connected components of the L-diagram, applies Cauchy–Schwarz

Load-bearing premise

The density theorem assumes that every integer between 0 and m(m−1)/2 occurs as the number of noninversions of some permutation in S_m; this fact is true, but the paper relies on it without proof or citation.

Editorial extensions

If this is right

  • For each n ≥ 4, the full set of achievable complexities is {0, 2, 3, ..., (n−1)(n−3)}; no other values occur.
  • The previously known absence of complexity 1 is now embedded in a complete integer-range statement.
  • The unique maximizer w0s_{n−1} has a one-box opposite Rothe diagram at (2, n−1), so the worst-case variety is determined by an explicitly simple permutation.
  • The reduction lemma gives a constructive way to lower complexity from d_max(n) by any desired amount, by inserting an arbitrary smaller permutation into the first n−2 positions.
  • The combinatorial bound shows that high complexity requires the L-diagram to concentrate its rows and columns into very few connected components, giving an obstruction to high complexity for permutations with scattered diagrams.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An analogous exhaustive-range question could be posed for other torus-invariant subvarieties of flag varieties; the formula here suggests the missing value 1 is special to these determinantal slices rather than a general phenomenon.
  • The proof of the density result depends on the fact that every integer from 0 to m(m−1)/2 occurs as the number of noninversions of some permutation in S_m; that fact is true via inversion numbers, but the paper neither states nor cites it.
  • One could try to construct an explicit family of permutations realizing every achievable complexity d uniformly, in the same spirit as the construction near the maximum, which would make the classification more constructive.
  • Because the maximum grows quadratically in n while only one small value is missing, the achievable set is eventually all integers in a quadratic range; for large n, torus complexity alone is a coarse invariant of these varieties.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies the complexity of certain torus-fixed affine subvarieties Y_w of matrix Schubert varieties X_w, where X_w ≅ Y_w × C^k with k maximal. The main results are: (i) for n ≥ 4, the maximal complexity among all w ∈ S_n is (n−1)(n−3), uniquely attained by w_0 s_{n−1} = [n, n−1, …, 3, 1, 2]; and (ii) every integer complexity d in {0, 2, 3, …, (n−1)(n−3)} is realized by some Y_w. The proofs use a graph-theoretic formula for complexity from Donten-Bury–Escobar–Portakal, an elementary bound on |L(w)|−|V(G_w)|, and a recursive construction that reduces the problem to smaller n. The argument is largely self-contained, though it relies on several published facts and contains some terse steps in Theorem 4.8.

Significance. If the results are correct, they give a complete classification of achievable complexities for the varieties Y_w, sharpening earlier work on toric matrix Schubert varieties (Escobar–Mészáros) and on the exclusion of complexity 1 (Donten-Bury–Escobar–Portakal). The main bound in Theorem 4.1 is clean and the equality analysis is convincing; the construction in Theorem 4.8 is explicit and elementary. The paper is a solid contribution to the combinatorial study of T-varieties, with results that are concrete and checkable. The proof of the maximum is a nice application of Cauchy–Schwarz to the associated bipartite graphs, and the uniqueness argument is carefully handled.

minor comments (4)
  1. [Section 4, proof of Theorem 4.8] The proof implicitly uses the fact that for every integer r between 0 and binom(m,2) there exists β ∈ S_m with |D^∘(β)| = r (equivalently, every inversion count occurs). This is true via Lehmer codes, but it is not stated or cited. Without this fact, the phrase 'we can achieve any complexity between d_max(k) and d_max(k) − (k−2)(k−3)/2' is not fully justified. Please add a sentence with a reference or a short proof.
  2. [Section 4, k = 5 case of Theorem 4.8] The complexity-4 construction w = [(n−5)+5, (n−5)+4, (n−5)+1, (n−5)+3, (n−5)+2, n−5, n−6, …, 1] is only verified for n = 5 in Example 3.9. For n > 5, the paper merely says 'see Example 3.9'. Please include the computation for general n (the same vertical-domino pattern applies), or state and prove that the complexity is 4 for all n ≥ 5.
  3. [Section 4, proof of Theorem 4.1, around Eq. (4.2)] The sentence 'Since L(w) is a skew diagram, this means L(w) has k connected components' is used to justify that the components of G_w match the components of L(w). This is true, but a one-line justification (each row and column of a skew diagram is contiguous, so graph connectivity coincides with box connectivity) would improve readability.
  4. [Section 4, Lemma 4.6] The definition of w in the proof is a bit terse, and the diagram in Figure 4.2 is informal. In particular, the footnote clarifying the shifted D^∘(β) is easy to miss. I suggest spelling out explicitly that w is the permutation whose one-line notation is [β_1+k, …, β_m+k, α_{m+1}, …, α_n] and that the boxes in D^∘(β) are shifted to the southwestern m×m block, as is already indicated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the classification is derived from independent combinatorial bounds and graph formulas, not from its conclusion.

full rationale

The paper's central claims—Theorem 4.1 (maximum complexity and unique maximizer) and Theorem 4.8 (attainment of all values except 1)—are derived from an explicit combinatorial formula for complexity, equation (3.6): d_w = |L'(w)| − |V(G_w)| + |C(G_w)|. This formula is not defined in terms of the target maximum or the target attainable set; it is a general expression in terms of diagrams and graphs. The bound d_w ≤ (n−1)(n−3) follows from independent inequalities (Cauchy–Schwarz, row/column counts, complete bipartite graph maximization), and the uniqueness argument uses equality conditions in those inequalities. Lemma 4.6 computes how complexity changes under a block construction by comparing the quantities |SW|, |D°|, |V(G)|, and |C(G)| before and after the construction; the result d_w = d_α − |D°(β)| is a computation, not an input. Theorem 4.8 then fills intervals using those constructions, and the exclusion of complexity 1 is cited to [5, Theorem 3.14], a published result of Donten-Bury–Escobar–Portakal rather than a restatement of this paper's theorem. The paper does rely on prior work coauthored by the first author, but those results are external, peer-reviewed, and have their own derivations; the present paper does not rename or repackage them as its own prediction. The only notable gap is that the proof of Theorem 4.8 assumes without stating or citing the standard fact that every integer between 0 and binom(m,2) occurs as |D°(β)| for some β ∈ S_m; this is a true combinatorial fact (e.g., via Lehmer codes) and is therefore an omitted justification, not a circular step. No fitted parameter is called a prediction, and no quantity is forced by self-definition.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters: the proof is a parameter-free combinatorial derivation built on published formulas. No new entities are postulated. The axioms are standard results in toric geometry, Schubert calculus, and permutation statistics; the only unstated one (surjectivity of inversion numbers) is standard and true.

assumptions (5)
  • standard math Fulton's theorem: dim X_w = n^2 − |D^o(w)|, ideal generated by rank conditions at Ess(w).
    Invoked as Theorem 2.7 (citing [7]); used to define Y_w and compute dim Y_w = |L'(w)|.
  • domain assumption Matrix Schubert varieties are normal.
    Cited to Knutson–Miller [8] after Thm 2.4.3; normality is required for the T-variety complexity framework.
  • domain assumption Complexity formula d_w = |L'(w)| − |V(G_w)| + |C(G_w)|.
    Equation (3.6) from [5]; combines the edge-cone dimension formula (Lemma 3.4) with (3.5). Central computational input.
  • standard math Every integer in [0, C(m,2)] occurs as the number of noninversions of some permutation in S_m.
    Used in Thm 4.8 to fill intervals; not stated in paper; follows from Lehmer-code/inversion-sequence surjectivity.
  • domain assumption The bipartite graph G_w has components matching the edge-connected components of the skew diagram L(w), with no isolated vertices.
    Used in Thm 4.1 to split L(w) into L_i with row/col counts r_i, c_i; standard for skew diagrams.

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Pith. "Pith review of Complexity of the Zero Set of a Matrix Schubert Ideal." pith.science (2026). https://pith.science/paper/LCGGYDEJ

@misc{pith2026251000131,
  author       = {Pith},
  title        = {Pith review of: Complexity of the Zero Set of a Matrix Schubert Ideal},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LCGGYDEJ}},
  note         = {Machine review of arXiv:2510.00131}
}
abstract

$T$-varieties are normal varieties equipped with an action of an algebraic torus $T$. When the action is effective, the complexity of a $T$-variety $X$ is $\dim(X)-\dim(T)$. Matrix Schubert varieties, introduced by Fulton in 1992, are $T$-varieties consisting of $n \times n$ matrices satisfying certain constraints on the ranks of their submatrices. In this paper, we focus on the complexity of certain torus-fixed affine subvarieties of matrix Schubert varieties. Concretely, given a matrix Schubert variety $\overline{X_{w}}$ where $w\in S_n$, we study the complexity of $Y_w$ obtained by the decomposition $\overline{X_{w}} = Y_{w} \times \mathbb{C}^{k}$ with $k$ as large as possible. Building up from results by Escobar and M\'{e}sz\'{a}ros and Donten-Bury, Escobar, and Portakal, we show that for a fixed $n$, the complexity of $Y_{w}$ with respect to this action can be any integer between $0$ and $(n-1)(n-3)$, except $1$.

Figures

Figures reproduced from arXiv: 2510.00131 by the authors.

Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 2.1
Figure 2.1. The submatrix Ma,b □ of M. This figure is adapted from [6]. The matrix Schubert variety associated to w ∈ Sn is the Zariski closure Xw := BwB ⊂ Cn×n. Fulton introduced matrix Schubert varieties in 1992 in his study of degeneracy loci of a map of flagged vector bundles [7]. He described the ideals defining matrix Schubert varieties combinatorially using Rothe diagrams. However, we will follow conventions set in [5] a… view at source ↗
Figure 2.2
Figure 2.2. The opposite Rothe diagram of the permutation [PITH_FULL_IMAGE:figures/full_fig_p004_2_2.png] view at source ↗
Figures from the paper (11 more)
Figure 2.3
Figure 2.3. Figure 2.3: The opposite Rothe diagram of the permutation [PITH_FULL_IMAGE:figures/full_fig_p004_2_3.png]
Figure 3
Figure 3. Figure 3: (e) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]
Figure 3.1
Figure 3.1. Figure 3.1: The opposite Rothe diagram, southwest diagram, [PITH_FULL_IMAGE:figures/full_fig_p005_3_1.png]
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]
Figure 3.2
Figure 3.2. Figure 3.2: The bipartite graph G3412 Following [5, pg. 841], Yw is a T-variety of complexity d with respect to the torus action T × T if and only if (3.5) dim(σw) = dim(Yw) − d = |L ′ (w)| − d. Let dw denote the complexity of the T-variety Yw with respect to the torus action T …
Figure 3
Figure 3. Figure 3: (d) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]
Figure 3.3
Figure 3.3. Figure 3.3: The opposite Rothe diagram, southwest diagram, [PITH_FULL_IMAGE:figures/full_fig_p006_3_3.png]
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 4.1
Figure 4.1. Figure 4.1: The opposite Rothe diagram, southwest diagram, [PITH_FULL_IMAGE:figures/full_fig_p008_4_1.png]
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 4.2
Figure 4.2. Figure 4.2: The opposite Rothe diagram of w = [β1 + k, . . . , βm + k, αm+1, . . . , αn]. Since the boxes in D◦ (α) are northeast of all boxes in the area labeled D◦ (β), we know that SW(w) = SW(α). Moreover, because D◦ (w) is the union of D◦ (α) and D◦ (β) (with the boxes in D◦…

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