On the weak and strong Lefschetz properties for initial ideals of determinantal ideals with respect to diagonal monomial orders

Hongmiao Yu

REVIEW 2 major objections 4 minor 32 references

On the weak and strong Lefschetz properties for initial ideals of determinantal ideals with respect to diagonal monomial orders

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For $t=\min\{m,n\}$, the initial ideal ring $R/\mathrm{in}(I_t)$ has the strong Lefschetz property; for smaller $t$, explicit matrix-size thresholds force the weak Lefschetz property to fail.

desk verdict The central Betti-number lemma is false as stated, and the paper's WLP-failure proof collapses; the SLP result for maximal minors may still survive repair, but the current manuscript is not publishable. read the letter →

arxiv 2506.05193 v2 pith:LBA6EYNQ submitted 2025-06-05 math.AC

Hongmiao Yu This is my paper

classification math.AC MSC 13C4013C70

keywords determinantalidealsinitialdiagonalmonomialordersweakLefschetzpropertystrongStanley–ReisnerringsGröbnerdegenerationsnonintersectingpaths

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 asks when the Lefschetz properties of a determinantal ring survive when the ideal is replaced by its square-free initial ideal under a diagonal monomial order. It establishes a complete positive result in the maximal-minor case: for $t=\min\{m,n\}$, the Stanley–Reisner ring $R/\mathrm{in}(I_t)$ has the strong Lefschetz property for every matrix size. In the non-maximal case $t<\min\{m,n\}$, it proves that $R/\mathrm{in}(I_t)$ fails even the weak Lefschetz property whenever the matrix has at least $16$ entries for $t=2$, at least $24$ entries for $t=3$, and at least $(t+1)(t+2)$ entries for $t\ge 4$. The failures are detected by a nonzero graded Betti number that blocks injectivity of every linear multiplication, combined with a Hilbert-function comparison that blocks surjectivity. As an application, these rings give counterexamples to the question whether square-free Gröbner degenerations preserve Lefschetz properties.

What carries the argument

The machinery is simplicial. Because the $t$-minors form a Gröbner basis under a diagonal monomial order, $\mathrm{in}(I_t)$ is a square-free monomial ideal, and its Stanley–Reisner complex $\Delta(t,m,n)$ has facets described as families of nonintersecting monotone paths in the $m\times n$ grid. The paper restricts $\Delta(t,m,n)$ to specially chosen vertex sets $V_a(t,m,n)$ of size $h+t-1$, giving subcomplexes $\Omega_a(t,m,n)$. Hochster's formula converts the Betti number $\beta_{h,h+t-1}$ into a sum of reduced homology groups of these restrictions, and Lemma 3.7 uses an induction on $t$ with links and a Cohen–Macaulayness argument to show each contributes at least one unit of $\tilde{H}_{t-2}$. The Lefschetz conclusion then comes from two counting mechanisms: Lemma 2.1 converts the nonzero homology into a non-injectivity statement, and Lemma 4.4's Hilbert-function inequality converts the same numerical setup into non-surjectivity.

What would settle it

Run the supplied computer algebra code (or an independent computation) for a small non-maximal case such as $t=3$, $m=n=5$ or $t=4$, $m=n=6$ and check whether $\beta_{h,h+t-1}(R/\mathrm{in}(I_t))\ge t$ holds; a value below $t$ would refute Theorem 1.2. A direct combinatorial test is to search the complex $\Delta(t,m,n)$ for a set of $t-1$ vertices of $\Gamma$ that is not a face; the smallest such failure would identify exactly where Lemma 3.7 breaks. Independently computing the reduced homology $\tilde{H}_{t-2}(\Omega_a(t,m,n);K)$ for all $a$ would either confirm the lower bound or expose the missing homology.

Watch

Extended reading notes

Core claim

The central claim is that the graded Betti number $\beta_{h,h+t-1}(R/\mathrm{in}(I_t))$ is at least $t$ for every $2\le t\le\min\{m,n\}$ with $t<\max\{m,n\}$, where $h=(m-t+1)(n-t+1)$ is the height of $\mathrm{in}(I_t)$. A nonzero entry in this Betti position means the Artinian reduction of $R/\mathrm{in}(I_t)$ has a socle element in degree $t-1$, and Lemma 2.1 turns that into the statement that multiplication by any linear form fails to be injective. Lemma 4.4 shows that when $F_t(m,n)=\binom{h+t-2}{t}-\binom{m}{t}\binom{n}{t}\ge 0$, the Hilbert function in degree $t$ is at most the Hilbert function in degree $t-1$, so the same multiplication map also fails to be surjective; together these force the weak Lefschetz property to fail. The thresholds in the Main Theorem are exactly the ranges where this Hilbert-function inequality holds, with one boundary pair checked by computer. In the complementary maximal-minor case the initial ideal has a $t$-linear resolution and its Artinian reduction is isomorphic to $K[Y_1,\ldots,Y_h]/(Y_1,\ldots,Y_h)^t$, a ring with the strong Lefschetz property, which proves the SLP for $R/\mathrm{in}(I_t)$.

Load-bearing premise

In the proof of Lemma 3.7 the paper asserts, with a reference to Remark 2.10, that every set of $t-1$ vertices of the deleted complex $\Gamma$ is a face; this amounts to claiming that the path complex is a matroid complex of rank $t-1$, a nontrivial combinatorial fact that is neither proved nor cited. If that fact fails, the shellability step and the Cohen–Macaulay conclusion for the $(t-2)$-skeleton collapse, and with them the Betti number lower bound $\beta_{h,h+t-1}\ge t$ and the WLP failure thresholds.

Editorial extensions

If this is right

For square matrices with $m=n\ge t+2$, the ring $R/I_t$ has the strong Lefschetz property while its square-free initial ideal $R/\mathrm{in}(I_t)$ fails the weak Lefschetz property, giving a negative answer to the open question about preservation under square-free Gröbner degenerations.
For $t=3$, $m=4$, $n=5$, $R/\mathrm{in}(I_3)$ has the weak Lefschetz property but fails the strong one, while $R/I_3$ has the strong Lefschetz property, so the failure can occur one level up.
For $t=3$, $m=4$, $n=6$, $R/\mathrm{in}(I_3)$ fails the weak Lefschetz property while $R/I_3$ has the strong Lefschetz property, extending the counterexample to rectangles.
For $t=2$ the threshold $mn\ge 16$ is sharp: computer calculations show $R/\mathrm{in}(I_2)$ has the strong Lefschetz property for all $mn\le 15$, and similarly $t=3$ has SLP for $m=n=4$ and WLP-without-SLP at $(4,5)$ and $(5,4)$.
If the equality $\mathrm{gin}(\mathrm{in}(I_t))=\mathrm{gin}(I_t)$ holds for $t=m-1=n-1$, then the bound in the Main Theorem is sharp for every $t$, as the paper raises in its closing question.

Reading between the lines

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

The unproved matroidal assertion in Lemma 3.7 — that every $t-1$ vertices of the deleted complex $\Gamma$ form a face — suggests that the real combinatorial core of the Betti lower bound is a matroid structure on the nonintersecting-path complex; a direct proof of that matroid property would place Theorem 1.2 on independent footing and could yield exact Betti numbers rather than the lower bound $t
The thresholds depend only on $mn$, not on the individual dimensions $m$ and $n$; a natural testable extension is whether WLP failure in the non-maximal case is exactly characterized by the same numerical condition $F_t(m,n)\ge 0$, or whether additional small exceptional rectangles behave differently.
The same two-step mechanism — a nonzero top Betti number blocking injectivity plus a Hilbert-function gap blocking surjectivity — is portable to other families of ideals whose initial ideals have a nonintersecting-path description, such as ladder determinantal ideals, where the path complexes are replaced by higher-dimensional analogues.
The contrast with the known sphere result for a different monomial order suggests that among square-free Gröbner degenerations the choice of monomial order controls Lefschetz behavior; comparing initial ideals under diagonal versus other orders might give a sharper criterion for when WLP is preserved.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: the paper takes on a real question—Murai's question and the SLP/WLP behavior of R/in(I_t)—and it has some genuinely useful pieces, but the main Betti-number lemma is false as stated. I checked the stress-test example and it works: for t=3, m=n=4, a=0, the vertices (1,2) and (2,1) in Γ=Ω_0(3,4,4)\{(4,4)} do not form a face. A path from (2,4) to (4,2) cannot pass through (2,1), given the allowed steps. So the assertion in Lemma 3.7 that every t−1 vertices form a face is wrong, and the shellability/Cohen-Macaulay argument built on it collapses. Theorem 1.2, Lemma 4.4, and the WLP-failure part of the Main Theorem are unproved as written.

Credit where it is earned: the paper is clearly organized, the diagonal-order framework is handled cleanly, and the Macaulay2 code is a real check on the edge cases. The t=2 computation (β_{h,h+1}=2 for m,n≥3) is solid and probably right. The SLP claim for maximal-minor initial ideals is plausible, and if a correct proof can be found, that would be a nice result. The negative answer to Murai's question may also be true, but it currently rests on the same broken lemma.

There is a second, smaller soft spot. Proposition 4.2 cites [26, Theorem 3.2] as an equivalence between having a t-linear resolution and a specific Hilbert-function value. As stated, that equivalence is missing hypotheses; ideals like (x,y)^2 in K[x,y,z] give the right Hilbert number for t=3 without having a 3-linear resolution. So the SLP direction needs scrutiny too.

Who is this for? A commutative algebra specialist interested in Lefschetz properties and Gröbner degenerations. The topic matters, and the paper shows the author knows the surrounding literature. But in its current form I would not send it to a serious referee. The counterexample is concrete, the central theorem is unsupported, and the fix is not a minor typo. The right move is to reject with a clear note telling the author to repair Lemma 3.7 or find an alternative argument, and to check the cited equivalence in Proposition 4.2. If those are fixed, the paper could be worth another look.

Referee Report

2 major / 4 minor

Summary. The paper studies the weak and strong Lefschetz properties for the Stanley–Reisner ring R/in(I_t), where I_t is the ideal of t-minors of an m×n matrix of indeterminates and in(I_t) is taken with respect to a diagonal monomial order. The main theorem claims that R/in(I_t) has the SLP when t=min{m,n}, and that R/in(I_t) fails the WLP for t<min{m,n} under the stated numerical conditions. The proof strategy is to bound a graded Betti number via Hochster's formula using a family of subcomplexes Ω_a(t,m,n), then to convert the Betti-number lower bound into failure of injectivity of multiplication by a linear form, while comparing Hilbert functions to obtain failure of surjectivity. The SLP case for maximal minors is approached through the linear resolution of in(I_t) and a cited characterization of ideals with linear resolutions.

Significance. If the main theorem is correct, the paper provides a substantial family of square-free Gröbner degenerations where Lefschetz properties fail, giving counterexamples to a question of Murai. The paper is largely self-contained, uses standard tools (Hochster's formula, the Wiebe–Murai transfer principle, Conca–Varbaro regularity, and Adiprasito's SLP theorem for spheres), and supplies Macaulay2 code for verification. There are no fitted parameters or circular reductions. However, two load-bearing arguments are not established as written: the key combinatorial assertion in Lemma 3.7 is false, and the cited equivalence in Proposition 4.2 is stated without the hypotheses needed to make it true. These gaps affect both the WLP-failure direction and the SLP direction of the main theorem.

major comments (2)

[Section 4, Proposition 4.2] The claim that 'Remark 2.10 implies that, for each t−1 vertices v1,...,v_{t−1} of Γ, the set {v1,...,v_{t−1}} is a face of Γ' is false. For t=3, m=n=4, a=0, we have V_0(3,4,4)={(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)} and F={(4,4)}, so (1,2) and (2,1) are vertices of Γ=Ω_0(3,4,4)\{(4,4)}. A face of Δ(3,4,4) must be contained in a family of nonintersecting paths from (1,4),(2,4) to (4,1),(4,2). Any path from (2,4) to (4,2) has all second coordinates at least 2, hence cannot contain (2,1); therefore {(1,2),(2,1)} is not a face of Γ. Consequently Γ^{≤t−2} is not the (t−2)-skeleton of a simplex, and the shellability/Cohen–Macaulay argument proving ~H_{t−3}(Γ)=0 collapses. Since Lemma 3.7 is the only route given to Theorem 1.2, and since Theorem 1.2 is used in Lemma 4.4, the WLP-failure part of the Main Theorem is not proved as written.
[Section 4, Proposition 4.2] The assertion that [26, Theorem 3.2] gives an equivalence 'J has a t-linear resolution if and only if HF(J,t)=binom(h+t−1,t)' is not valid under the hypotheses stated in the paper. For example, J=(x^2,y^2)⊂K[x,y] has a 2-linear resolution, and K[x,y]/J is Artinian and Cohen–Macaulay, but HF(J,2)=2 while binom(3,2)=3. Thus the forward direction used in the proof is false without additional hypotheses. The conclusion J=(Y_1,...,Y_h)_t, which is essential for the SLP argument for maximal minors, therefore does not follow from the cited theorem as stated. The author should either give the precise hypotheses of the cited result and verify them, or supply a different proof of the structure of the Artinian reduction.

minor comments (4)

[Definition 2.6] The word 'filed' should be 'field'.
[Lemma 3.7] Even if the face claim were true, the citation to Remark 2.10 would be incomplete: Remark 2.10 only describes the facets of Δ(t,m,n) and does not directly assert that arbitrary sets of t−1 vertices lie in a common face, so the inference needs a separate proof.
[Section 5] The Macaulay2 code is useful and reproducible, but the printed code has formatting and line-break issues that make it difficult to verify directly from the PDF; the GitHub link is therefore important and should be prominently referenced.
[Remark 4.5] The use of Rubey–Stump's result in [27] is qualified by a footnote explaining that the proof was not formally published; this qualification should be stated in the main text as well, since the SLP of R/I_t depends on it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation relies on standard external results and explicit combinatorial and Hilbert-function arguments, with no fitted input renamed as prediction.

full rationale

The paper does not exhibit any of the enumerated circularity patterns. Its main results are derived from standard external tools (Hochster's formula, Wiebe–Murai transfer, Conca–Varbaro regularity/Cohen–Macaulay results, Narasimhan's Gröbner basis theorem, Bruns–Conca linear-resolution facts, and Adiprasito's SLP theorem for simplicial spheres), and none of these citations are self-citations of the author. There are no fitted constants, no parameters calibrated to a subset of data and then presented as predictions, and no model output that is equivalent to its input by construction. The Betti-number argument in Section 3 is an internal induction based on Lemma 2.15 and Lemma 3.7; the latter's disputed combinatorial assertion is a matter of mathematical correctness, not of circularity, since it is not obtained by assuming the conclusion. Proposition 4.2 and Theorem 4.3 similarly reduce to standard ring-theoretic equivalences and explicit inequalities plus Macaulay2 verifications. The manuscript itself even includes a limitation note that the Rubey–Stump proof was not formally published, which is transparency about external support rather than a circular reliance. Therefore no circular step can be quoted, and the appropriate finding is an honest non-finding with score 0.

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

The central claim rests on standard external results (Groebner basis theorem, Hochster's formula, Conca-Varbaro) plus one unproved combinatorial assertion about the path complex Delta(t,m,n) acting like a matroid complex; no free parameters are fitted and no new entities are postulated.

assumptions (5)

standard math Narasimhan's theorem: the t-minors form a Groebner basis of I_t with respect to any diagonal monomial order, so in(I_t) is square-free.
Invoked in Section 2.2 to identify R/in(I_t) with a Stanley-Reisner ring; cited to [23], [5], [9], [30].
standard math Hochster's formula computing graded Betti numbers of Stanley-Reisner ideals from reduced homology of restrictions.
Used throughout Section 3 to turn the Betti number bound into a homology computation.
domain assumption For each t-1 vertices v_1,...,v_{t-1} of Gamma, the set {v_1,...,v_{t-1}} is a face of Gamma (equivalently, Delta(t,m,n) has a matroid complex of rank t-1).
Asserted in the proof of Lemma 3.7 as an implication of Remark 2.10, but not proved; it is load-bearing for Theorem 1.2.
standard math Conca-Varbaro's result that square-free initial ideals preserve regularity and Cohen-Macaulayness.
Used in Remark 4.1 and in the SLP proof for maximal minors; cited to [12].
domain assumption Soll-Welker conjecture, proved by Rubey-Stump, that certain monomial order initial ideals define simplicial spheres; Adiprasito's theorem that Stanley-Reisner rings of simplicial spheres have SLP.
Used in Remark 4.5 to conclude R/I_t has SLP; the Rubey-Stump proof is only on arXiv (not formally published).

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Pith. "Pith review of On the weak and strong Lefschetz properties for initial ideals of determinantal ideals with respect to diagonal monomial orders." pith.science (2026). https://pith.science/paper/LBA6EYNQ

@misc{pith2026250605193,
  author       = {Pith},
  title        = {Pith review of: On the weak and strong Lefschetz properties for initial ideals of determinantal ideals with respect to diagonal monomial orders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LBA6EYNQ}},
  note         = {Machine review of arXiv:2506.05193}
}

abstract

We study the weak and strong Lefschetz properties for $R/\mathrm{in}(I_t)$, where $I_t$ is the ideal of a polynomial ring $R$ generated by the $t$-minors of an $m\times n$ matrix of indeterminates, and $\mathrm{in}(I_t)$ denotes the initial ideal of $I_t$ with respect to a diagonal monomial order. We show that when $I_t$ is generated by maximal minors (that is, $t=\mathrm{min}\{m,n\}$), the ring $R/\mathrm{in}(I_t)$ has the strong Lefschetz property for all $m$, $n$. In contrast, for $t<\mathrm{min}\{m,n\}$, we provide a bound such that $R/\mathrm{in}(I_t)$ fails to satisfy the weak Lefschetz property whenever the product $mn$ exceeds this bound. As an application, we present counterexamples that provide a negative answer to a question posed by Murai regarding the preservation of Lefschetz properties under square-free Gr\"obner degenerations.

Figures

Figures reproduced from arXiv: 2506.05193 by the authors.

**Figure 1.** The set Va(t, m, n) Definition 2.12. For each integer 0 ≤ a ≤ t − 1, Ωa(t, m, n) = {σ ∈ ∆(t, m, n) | σ ⊆ Va(t, m, n)} is the simplicial complex defined as the restriction of ∆(t, m, n) to the set Va(t, m, n). Example 2.13. If t = 2, m = n = 3, then the vertex set of Ω0(2, 3, 3) is V0(2, 3, 3) = {(1, 1),(1, 2),(2, 1),(2, 2),(3, 3)} [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** The set V0(2, 3, 3) Moreover, by Remark 2.10, the faces of Ω0(2, 3, 3) are the restrictions to the set V0(2, 3, 3) of the chains from (1, 3) to (3, 1). For instance, since {(1, 3),(1, 2),(1, 1),(2, 1),(3, 1)} is a chain from (1, 3) to (3, 1), it follows that {(1, 2),(1, 1),(2, 1)} is a face of Ω0(2, 3, 3). Furthermore, the chain {(1, 2),(1, 1),(2, 1)} is maximal in V0(2, 3, 3), and so it is a facet of Ω0(2, 3, 3). S… view at source ↗

**Figure 3.** Ω0(2, 3, 3) Example 2.14. If t = 3, m = 4, n = 5 and a = 1, then the vertex set of Ω1(3, 4, 5) is V1(3, 4, 5) = {(1, 1),(2, 2),(2, 3),(2, 4),(3, 2),(3, 3),(3, 4),(4, 5)}. As shown in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The restriction to the set V1(3, 4, 5) of the family of nonintersecting paths F The facets of Ω1(3, 4, 5) are {(1, 1),(2, 2),(2, 3),(2, 4),(3, 2)}, {(1, 1),(2, 4), (3, 2),(3, 3),(3, 4)}, {(1, 1),(2, 3),(2, 4),(3, 2),(3, 3)}, {(1, 1),(4, 5)}, {(2, 2), (2, 3),(2, 4),(3, …

**Figure 5.** Figure 5: V0(2, m, n) (left) and V1(2, m, n) (right) By Remark 2.10, F = {f} is a facet of Ωa(2, m, n), and q ≤ p for each q ∈ Va(2, m, n) \ F. It follows that p ∈ \ G is a facet of Ωa(2,m,n) G̸=F G. Moreover, ⟨F⟩ ∪ Ωa(2, m, n) \ F = Ωa(2, m, n) and ⟨F⟩ ∩ Ωa(2, m, n) \ F = {…

**Figure 6.** Figure 6: Possible vertex region of Ω. We prove that if dimK H˜ 0(Ω; K) ≥ 1 then Ω = Ωa(2, m, n) for some a ∈ {0, 1} through the following sequence of steps: Step 1: Show that {(i, j) | k ≤ i ≤ m, 1 ≤ j ≤ f} ∩ Vert(Ω) = ∅. Step 2: Show that s = 1, f = l−1, and the vertices of Ω …

**Figure 11.** Figure 11: s = 2, n = l and f = 1 In both cases, (2, n − 1) ∈ C ∩ D ∩ U holds by applying the assumption m, n ≥ 3 and using f ≤ l − 2 ≤ n − 2, 2 = s ≤ k − 1 < m again. Moreover, (i, j) ≤ (k − 1, 1) for all (i, j) ∈ C ∩ U and (i, j) ≤ (m, f + 1) for all (i, j) ∈ D∩U. In particula…

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