{"id":"eca21f09-1868-4dcc-92bd-66c0d1eb07c2","arxiv_id":"2506.05193","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For initial ideals of determinantal ideals with respect to diagonal monomial orders, the author proves SLP for maximal minors and WLP failure for non-maximal minors when mn is large enough.","lead":"This paper studies when the initial ideal of the ideal of t-minors of a generic matrix, with respect to a diagonal monomial order, has the weak or strong Lefschetz property. It proves the strong Lefschetz property in the maximal minor case and gives explicit bounds where the weak Lefschetz property fails, yielding counterexamples to a question posed by Murai.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.7's key combinatorial assertion is false: for t=3, m=n=4, a=0, the vertices (1,2) and (2,1) of Γ=Ω_0(3,4,4)\\(4,4) do not form a face, so the proof of Theorem 1.2 collapses.","rationale":"The reader's conditional verdict is appropriate, and the flagged assumption is exactly where the argument breaks. I independently verified that the Lemma 3.7 assertion is not merely unproved but false: the pair {(1,2),(2,1)} in Ω_0(3,4,4)\\{(4,4)} cannot be extended to any family of two nonintersecting paths with the prescribed endpoints. This invalidates the stated proof of Theorem 1.2 and hence the WLP-failure results that depend on it. I do not claim the main theorem is false; the Betti bound may be recoverable by a different argument, and the paper has credible independent ingredients: Narasimhan's Gröbner basis theorem, Hochster's formula, the Macaulay2 computations in Section 5, and the cited SLP results of Adiprasito and of Rubey–Stump for R/I_t. However, Proposition 4.2 also cites [26, Theorem 3.2] in a way that appears to assert a false equivalence, so a careful revision should address both gaps. Unless the author supplies a corrected proof of Lemma 3.7 and a precise citation or derivation for Proposition 4.2, the paper should not be accepted as is; this is consistent with the reader's CONDITIONAL verdict.","tokens_in":22517,"tokens_out":16695,"duration_ms":188590,"concrete_test":"Construct Ω_0(3,4,4) from the facet description in Remark 2.10: facets are the restrictions to V0={(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)} of the nonintersecting path families from (1,4),(2,4) to (4,1),(4,2). Delete vertex (4,4) and test whether the 2-element set {(1,2),(2,1)} is a face of the resulting complex. It will fail, thereby disproving the assertion used in Lemma 3.7. A complementary check is to compute β_{4,6}(R/in(I_3)) for m=n=4 with Macaulay2: Theorem 1.2 predicts the value is at least 3, and this computation determines whether the Betti bound itself survives or needs an alternative proof.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 3.7 asserts that for Γ=Ω_a(t,m,n)\\F, every set of t−1 vertices of Γ is a face, claiming this follows from Remark 2.10. This assertion is false. Take t=3, m=n=4, a=0. Then V0={(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. No two nonintersecting paths from (1,4),(2,4) to (4,1),(4,2) can contain them respectively: a path from (1,4) to (4,1) through (1,2) must pass through (2,2) and then through one of (2,1), (3,2), or (1,1); a path from (2,4) to (4,2) through (2,1) must pass through (2,2),(2,1),(3,1) and then through (4,1) or (3,2),(4,2); every combination intersects. Hence {(1,2),(2,1)} is not a face, and the inference that Γ^{≤t−2} is the (t−2)-skeleton of a simplex is invalid. Since this is the only argument for the vanishing of H_{t−3}(Γ), Lemma 3.7, and therefore Theorem 1.2, Lemma 4.4, and the WLP-failure part of the Main Theorem, are not proved as written. Separately, Proposition 4.2's appeal to [26, Theorem 3.2] appears to state an equivalence that is false for, e.g., J=(x^2)⊂K[x,y], so the SLP direction also needs scrutiny, but the Lemma 3.7 failure is the primary obstruction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22907,"tokens_out":10413,"duration_ms":122503,"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":[{"comment":"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":"Section 4, Proposition 4.2"},{"comment":"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.","section":"Section 4, Proposition 4.2"}],"minor_comments":[{"comment":"The word 'filed' should be 'field'.","section":"Definition 2.6"},{"comment":"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":"Lemma 3.7"},{"comment":"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.","section":"Section 5"},{"comment":"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.","section":"Remark 4.5"}],"recommendation":"major_revision","confidential_remarks":"The false combinatorial claim in Lemma 3.7 is the most serious problem; the counterexample with t=3, m=n=4 shows that the proof of Theorem 1.2 cannot be repaired by a minor edit. The statement about [26, Theorem 3.2] in Proposition 4.2 also needs precise verification. I would encourage the editor to send the manuscript back for a substantial revision rather than reject outright, because the main theorems may still be true and the paper contains useful ideas and reproducible code."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nCredit 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.\n\nThere 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.\n\nWho 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.","headline":"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.","tokens_in":23436,"tokens_out":5629,"would_cite":false,"duration_ms":65782,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13C40","13C70"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["determinantal ideals","initial ideals","diagonal monomial orders","weak Lefschetz property","strong Lefschetz property","Stanley–Reisner rings","Gröbner degenerations","nonintersecting paths"],"falsifier":"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.","tokens_in":22282,"feed_emoji":"🧮","tokens_out":24472,"duration_ms":232907,"temperature":0.7,"pith_summary":"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.","feed_headline":"Maximal-minor initial ideals keep the strong Lefschetz property","feed_subtitle":"Smaller minors lose the weak Lefschetz property past explicit size thresholds. Square-free degenerations can break it.","key_machinery":"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.","core_discovery":"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 $R/\\mathrm{in}(I_t)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the description of the facets of $\\Delta(t,m,n)$ as families of nonintersecting paths, the Cohen–Macaulayness of $R/\\mathrm{in}(I_t)$, and the dimension formulas used throughout.","marker":"[7]"},{"why":"Shows the $t$-minors form a Gröbner basis for diagonal monomial orders, so $\\mathrm{in}(I_t)$ is square-free and defines the simplicial complex $\\Delta(t,m,n)$.","marker":"[23]"},{"why":"Hochster's formula, used to express $\\beta_{h,h+t-1}$ as a sum of reduced homology groups of restrictions of $\\Delta(t,m,n)$.","marker":"[15]"},{"why":"Provides the background results on simplicial homology, links, shellability, and Cohen–Macaulayness used in Lemma 3.7 and Lemma 2.1.","marker":"[8]"},{"why":"Proves that skeleta of shellable complexes are shellable, a step in showing the $(t-2)$-skeleton of $\\Gamma$ is Cohen–Macaulay in Lemma 3.7.","marker":"[4]"},{"why":"A Cohen–Macaulay criterion for simplicial complexes, used to obtain the homology vanishing that powers the induction in Lemma 3.7.","marker":"[25]"},{"why":"Gives the $t$-linear resolution of the determinantal ideal $I_t$, the input for the SLP proof in Proposition 4.2.","marker":"[6]"},{"why":"Shows regularity is preserved for square-free Gröbner degenerations, allowing the $t$-linear resolution to pass from $I_t$ to $\\mathrm{in}(I_t)$.","marker":"[12]"},{"why":"Completes the proof that the simplicial complex associated to the monomial order constructed in [28] is a sphere, so its Stanley–Reisner ring is a sphere ring.","marker":"[27]"},{"why":"Proves the strong Lefschetz property for Stanley–Reisner rings of simplicial spheres, giving the SLP of $R/I_t$ in the application.","marker":"[1]"}],"fun_headline_variants":["Maximal-minor initial ideals retain strong Lefschetz property","For t < min(m,n), initial ideals lose weak Lefschetz past a bound","Square-free Grobner degeneration can destroy Lefschetz properties","Maximal minors keep SLP; smaller minors fail WLP past threshold"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Maximal-minor initial ideals retain strong Lefschetz property","For t < min(m,n), initial ideals lose weak Lefschetz past a bound","Square-free Grobner degeneration can destroy Lefschetz properties","Maximal minors keep SLP; smaller minors fail WLP past threshold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001012,"raw_usage":{"total_tokens":4340,"prompt_tokens":1078,"completion_tokens":3262,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":694,"completion_tokens_details":{"reasoning_tokens":3184}},"tokens_in":694,"tokens_out":3262,"duration_ms":30798,"temperature":1.0,"reasoning_tokens":3184,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:28:24.886562+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Determinants, Gr¨ obner Bases and Cohomology","cited_arxiv_id":null,"evidence_quote":"Supplies the description of the facets of $\\Delta(t,m,n)$ as families of nonintersecting paths, the Cohen–Macaulayness of $R/\\mathrm{in}(I_t)$, and the dimension formulas used throughout."},{"cited_title":"The irreducibility of ladder determinantal varieties","cited_arxiv_id":null,"evidence_quote":"Shows the $t$-minors form a Gröbner basis for diagonal monomial orders, so $\\mathrm{in}(I_t)$ is square-free and defines the simplicial complex $\\Delta(t,m,n)$."},{"cited_title":"Cohen-macaulay rings, combinatorics, and simplicial complexes","cited_arxiv_id":null,"evidence_quote":"Hochster's formula, used to express $\\beta_{h,h+t-1}$ as a sum of reduced homology groups of restrictions of $\\Delta(t,m,n)$."},{"cited_title":"Cambridge Studies in Advanced Mathematics","cited_arxiv_id":null,"evidence_quote":"Provides the background results on simplicial homology, links, shellability, and Cohen–Macaulayness used in Lemma 3.7 and Lemma 2.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that skeleta of shellable complexes are shellable, a step in showing the $(t-2)$-skeleton of $\\Gamma$ is Cohen–Macaulay in Lemma 3.7."},{"cited_title":"Cohen-macaulay quotients of polynomial rings","cited_arxiv_id":null,"evidence_quote":"A Cohen–Macaulay criterion for simplicial complexes, used to obtain the homology vanishing that powers the induction in Lemma 3.7."},{"cited_title":"Products of borel fixed ideals of maximal minors","cited_arxiv_id":null,"evidence_quote":"Gives the $t$-linear resolution of the determinantal ideal $I_t$, the input for the SLP proof in Proposition 4.2."},{"cited_title":"Square-free gr¨ obner degenerations","cited_arxiv_id":null,"evidence_quote":"Shows regularity is preserved for square-free Gröbner degenerations, allowing the $t$-linear resolution to pass from $I_t$ to $\\mathrm{in}(I_t)$."},{"cited_title":"Crossings and nestings in set partitions of classical types","cited_arxiv_id":"0904.1097","evidence_quote":"Completes the proof that the simplicial complex associated to the monomial order constructed in [28] is a sphere, so its Stanley–Reisner ring is a sphere ring."}],"review_version":1}