{"id":"6dfeb448-96d9-459e-84e2-c4884cb894e6","arxiv_id":"2506.20153","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors provide path-based sufficient conditions for a Kazhdan-Lusztig ideal to be inhomogeneous, plus explicit examples outside Neye's standard-homogeneous class.","lead":"This paper studies when certain algebraic objects called Kazhdan-Lusztig ideals have a special homogeneous structure, and it proposes a combinatorial test for when they do not. It also gives explicit examples showing that a known sufficient condition is not necessary; a generalist would read it to see how permutation patterns control the algebraic complexity of Schubert varieties.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.0.1, the algebraic foundation of the §7.1 inhomogeneity test, is not established: Case 4 (c=1) rests on an unjustified cancellation claim, and Remark 4.2.0.2 is asserted without proof.","rationale":"The reader correctly identifies Lemma 5.0.1 as the load-bearing algebraic assumption. My reading of Section 5 confirms that the proof has a genuine gap: the Case 4 argument does not account for products of nonconstant parts of g_1 with other homogeneous components of f_1, which can have the same degree as the chosen component and therefore can participate in cancellation. This is not a mere stylistic issue; it is exactly the step needed to guarantee that some component is covered by monomials of other generators or components. The essential-set reduction in Remark 4.2.0.2 is also unproved, and both the algorithm and the explicit family in Example 8.0.2 depend on it. That said, I do not claim Lemma 5.0.1 is false; the lemma is plausible and may admit a correct degree-filtration proof. The paper contains useful partial results, a checkable example family, and an honest statement of what remains incomplete. The conditional verdict is appropriate: the central sufficient-condition algorithm should not be fully accepted until Lemma 5.0.1 is rigorously proved and the essential-set reduction is justified. My concern therefore leaves the reader's verdict unchanged.","tokens_in":25217,"tokens_out":29311,"duration_ms":321349,"concrete_test":"Re-derive Lemma 5.0.1 by induction on the lowest degree of a homogeneous component of f_1: take the degree-d part of a syzygy h_{1d} = Σ g_i f_i and show that if the coefficient of f_1 is not 1 then h_{1d} is covered by monomials of f_i for i≠1, while if it is 1, the residual cancellation at degree d must be controlled by the next component, whose monomials are divisible by monomials of h_{1d} or of f_i (i≠1). If this filtration argument cannot be completed without Case 4's assertion, the lemma is unproven. A useful secondary check is to compute the reduced Gröbner basis of Example 8.0.2 for n=6 and confirm that no additional defining minors appear beyond the two claimed generators; if additional generators appear, the example's inhomogeneity argument does not follow from the reduced set.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 7.1 declares I_{v,w} inhomogeneous exactly by the contrapositive of Lemma 5.0.1, so the lemma's validity is load-bearing. The proof of Lemma 5.0.1 is incomplete. In Case 4, subcase c=1, the authors need to show that some component h_{1k} is obtained from Σ_{i≠1} g_i f_i plus (g_1−1) times the remaining components. They justify this by saying that a nonconstant monomial times h_{1k} cannot give a term of h_{1k}, and that the other components h_{1j} have different degrees. That is insufficient: products of the nonconstant part of g_1 with h_{1j} (j≠k) or with h_{11} can have degree equal to deg h_{1k} and can cancel monomials of h_{1k}. The proof does not analyze the degree filtration of the syzygy, so the claimed existence of a component all of whose monomials are divisible by monomials of other generators/components is not established. Without Lemma 5.0.1, the contrapositive test in §7.1 has no proven foundation. Separately, the algorithm and Example 8.0.2 rely on the essential-set reduction in Remark 4.2.0.2, which is asserted without proof; if the reduced set is not a generating set, Lemma 5.0.1 cannot be applied to it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the homogeneity of Kazhdan-Lusztig ideals I_{v,w} for permutations v,w in S_n. The main contribution is an algorithmic procedure (Section 7.1) that gives sufficient conditions for I_{v,w} to be inhomogeneous. The algorithm combines: (i) a reduction to an 'essential set' of defining minors (Remark 4.2.0.2); (ii) a general algebraic lemma (Lemma 5.0.1) stating that a homogeneous ideal generated by polynomials containing an inhomogeneous generator must have a homogeneous component of that generator whose monomials are all divisible by monomials of other generators or other components; (iii) combinatorial criteria, developed in Section 6, for when a defining minor of Z^{(v)} is singular, when its determinant is inhomogeneous, and when one monomial of a determinant divides another. The paper also gives examples, including an infinite family in S_n (n ≥ 6) claimed to be inhomogeneous (Example 8.0.2), and contrasts these with Neye's sufficient conditions for standard homogeneity. Finally, Section 9 sketches a 'mutation' procedure intended to yield necessary and sufficient conditions for homogeneity, but the authors explicitly state that no concrete classification has been obtained from it.","tokens_in":25495,"tokens_out":5945,"duration_ms":61293,"significance":"If the main claims were fully established, the paper would provide the first general sufficient conditions for inhomogeneity of Kazhdan-Lusztig ideals that go beyond small cases, and would connect the homogeneity problem to explicit path combinatorics in the matrix Z^{(v)}. The paper is honest about its limitations: the mutation procedure of Section 9 is unfinished, and the authors do not claim a complete classification. The most valuable standalone result is Theorem 6.1.4, which gives a clean necessary and sufficient condition for a single defining minor to have inhomogeneous determinant. The paper also makes good use of external benchmarks from Neye [Ney23] and Woo-Yong [WY12], and there is no apparent circularity in the main criteria. However, the central inhomogeneity test in Section 7.1 rests on Lemma 5.0.1, whose proof contains gaps, and on an essential-set reduction that is asserted without proof. These issues are load-bearing: without them, the algorithm's final declaration of inhomogeneity has no proven foundation.","major_comments":[{"comment":"The proof of Lemma 5.0.1 in Case 2 asserts that 'Since h13,...,h1n1 are distinct from h12, they cannot contribute to h12.' This is not justified: products of the polynomials g_i with the components h_{1j} (j ≠ 2) can have terms of the same degree as terms of h_{12}, and cancellation between different homogeneous components can occur. The proof needs an analysis of the degree filtration of the relation h_{12} = -Σ_{i≠1} g_i f_i + (h_{13}+...+h_{1n1}) to show that some component h_{1j} must have every monomial divisible by a monomial of another generator or component. Without this, Lemma 5.0.1 is not established, and the contrapositive test in Section 7.1 collapses.","section":"Section 5, Lemma 5.0.1, Case 2"},{"comment":"In the subcase c = 1, the proof claims that a nonconstant monomial multiplied by h_{1k} cannot give a term of h_{1k}, and that the other components h_{1j} have different degrees, 'therefore −h_{1k} is obtained from the part Σ_{i=2}^n g_i f_i + (g_1−1)(h_{11}+...+h_{1,k−1}+h_{1,k+1}+...+h_{1n1})'. This does not follow: products of the nonconstant part of g_1 with h_{1j} for j ≠ k (or with h_{11}) can have degree equal to deg h_{1k} and can cancel monomials of h_{1k}. The proof must analyze the degree filtration of the syzygy, not merely assert the absence of such contributions. This gap is load-bearing because Section 7.1 uses the contrapositive of Lemma 5.0.1 as the final inhomogeneity test.","section":"Section 5, Lemma 5.0.1, Case 4(a)"},{"comment":"The reduction to the essential set is stated as 'One can prove a lemma similar to lemma 3.10 in [Ful92]' but no proof is given, and the translation from Fulton's northwest convention to the southwest convention is not carried out. The algorithm in Section 7.1 and the generator computation in Example 8.0.2 ('one can check that the only relevant minors...') rely on this reduction. Without a proof or a precise statement with the adaptation to Z^{(v)} and the defining minors in the southwest convention, the set of generators used in the final divisibility test is not known to generate I_{v,w}.","section":"Remark 4.2.0.2"},{"comment":"The example concludes that I_{v,w} = <f_1,f_2> and that this ideal is inhomogeneous by Lemma 5.0.1. This conclusion depends on both the unproved essential-set reduction (Remark 4.2.0.2) and on the unproved Lemma 5.0.1. In addition, the verification that no other defining minors contribute is not shown for general n; the displayed matrices and the claim that minors for t ≥ 5 are excluded need a careful case-by-case check. There is also a typo in the definition of w: 'w(30) = n−1' should be 'w(3) = n−1'.","section":"Example 8.0.2"}],"minor_comments":[{"comment":"The set [n] is defined as '{1.2,...,n}' with a period instead of a comma; please fix this typo, along with the various similar typographical errors throughout the paper.","section":"Section 1, page 2"},{"comment":"The sentence 'as g (a nonzero path) is supposed to divide the path g' should read 'divide the path f'; otherwise the proof is confusing.","section":"Section 6.1, Observation 2 proof"},{"comment":"The phrase 'A has one row or column zero' should be stated as 'A has a zero row or a zero column'. The conditions (3) and (4) use notation such as 'i_{s+1}' when s = p−1 without defining a boundary case, and the parenthetical alternatives make the statement hard to parse; please rewrite these conditions more explicitly.","section":"Lemma 6.1.2"},{"comment":"The determinant computations for the submatrices Z^{(v)}_{22} and Z^{(v)}_{33} are not fully explained; in particular, the row and column indexing relative to Z^{(v)} should be specified so that the displayed minors can be checked directly.","section":"Example 8.0.1"},{"comment":"The algorithm is described as a list of steps, but there is no worked example of the entire procedure for a small pair (v,w) beyond the partial illustration in Example 8.0.2. A complete example for n = 6 would help the reader verify each step.","section":"Section 7.1"},{"comment":"The mutation procedure is admittedly incomplete ('we have not yet been able to deduce any concrete set of necessary as well as sufficient conditions'). This is acceptable as a sketch, but the paper should clearly label the theorem in this section as conditional on the termination of the procedure and on the checking steps in Section 9.2.1.","section":"Section 9"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a meaningful problem in Schubert geometry/combinatorics and contains a useful criterion (Theorem 6.1.4) for inhomogeneity of a single defining minor. The main obstacle to acceptance is not lack of novelty but lack of proof of the two load-bearing ingredients: Lemma 5.0.1 and the essential-set reduction. If the authors can repair the proof of Lemma 5.0.1 (or replace it with a correct but weaker lemma that still supports the contrapositive test) and provide a proof of the essential-set reduction in their convention, the paper would be substantially stronger. I would also encourage them to verify Example 8.0.2 with a complete calculation for general n, since the current 'one can check' leaves the example partly unsupported. The paper's honesty about the incomplete mutation section is commendable, but the central claims should be made fully rigorous before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper has a genuinely useful new path criterion and some good examples, but the main inhomogeneity test rests on Lemma 5.0.1, which is not established as written. I would still send it to a referee, because the useful parts are separable and the problem is real.\n\nWhat's new and good: Theorem 6.1.4 gives a path/column condition for a minor of Z(v) to have inhomogeneous determinant, and the supporting lemmas in Section 6 are combinatorial and independently checkable. Example 8.0.2 constructs an explicit infinite family of pairs (v,w) outside Neye's sufficient conditions; even if the proof needs fixing, this is the right kind of concrete evidence. The early reductions are also reasonable: discarding w = reverse permutation in Lemma 3.1.1 and the whole-ring criterion in Theorem 3.1.1 look correct and useful. There are no self-citation chains or fitted parameters; the criteria are derived from the matrix structure, not tuned to benchmark cases.\n\nWhere it breaks down: Lemma 5.0.1 is load-bearing, and the proof has real gaps. Case 2 asserts that distinct homogeneous components h_{12}, ..., h_{1n_1} cannot contribute to h_{12} without a degree or cancellation argument. Case 4(c=1) says a nonconstant monomial times h_{1k} cannot produce terms of h_{1k} and then concludes -h_{1k} must come from the rest; but products of the nonconstant part of g_1 with other h_{1j}'s can have the same degree as h_{1k} and cancel part of it. The stress-test note is on target. Since Section 7.1 declares inhomogeneity exactly by the contrapositive of this lemma, the algorithm's final step is unproven. Example 8.0.2's inhomogeneity claim is also justified through Lemma 5.0.1, so it inherits the gap, although the underlying polynomials look as though they should give a true example.\n\nAlso, Remark 4.2.0.2 says 'one can prove' a Fulton-style essential-set reduction but gives no proof, and the algorithm depends on it to discard redundant minors. Section 9 is explicitly unfinished and should not be treated as a contribution. There are typos (e.g., 'w(30)' in Example 8.0.2) and the text needs an editorial pass.\n\nWho benefits: people working on Schubert patch ideals, determinantal ideals, or standard homogeneity. The path criterion and the examples are worth thinking about even while the main theorem is in doubt.\n\nRecommendation: send to peer review, not desk reject. A competent referee can separate the solid combinatorial criteria from the unproven lemma. The authors should be asked to prove Lemma 5.0.1 properly or find a counterexample, to prove the essential-set reduction, and to downgrade the algorithmic claims accordingly. Acceptance as-is would be premature.","headline":"A serious but currently unproven inhomogeneity test for KL ideals; the path criterion and the explicit examples are worth a referee's time.","tokens_in":26041,"tokens_out":4123,"would_cite":true,"duration_ms":44183,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E14","05E40","14N10","13C70"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper gives a path-based, algorithmic sufficient condition for a Kazhdan-Lusztig ideal to be inhomogeneous, and sketches a mutation procedure aimed at necessary and sufficient conditions for standard homogeneity.","keywords":["Kazhdan-Lusztig ideal","standard homogeneous","inhomogeneous ideal","Schubert variety","determinantal ideal","path","essential set","rank matrix"],"falsifier":"The lemma would be refuted by a concrete homogeneous ideal $I=\\langle f_1,\\dots,f_m\\rangle$ over $\\mathbb{C}$ with an inhomogeneous generator $f_i$ such that every homogeneous component $h_{ij}$ has at least one monomial divisible by no monomial of any other generator and by no monomial of any other component of $f_i$; checking the \"cannot contribute\" step in Case 2 of the lemma's proof on a small example would settle the point.","tokens_in":24958,"feed_emoji":"🧩","tokens_out":8983,"duration_ms":85967,"temperature":0.7,"pith_summary":"The paper studies Kazhdan-Lusztig ideals, determinantal ideals that cut out affine patches of Schubert varieties at torus-fixed points, indexed by pairs of permutations $(v,w)$. Its target is the homogeneity problem: deciding when such an ideal admits a generating set made of homogeneous polynomials, a property the authors call \"standard homogeneous\". The main claim is a sufficient condition for inhomogeneity, implemented as a finite algorithm: discard trivial and redundant defining minors, filter out singular minors by path criteria, identify minors whose determinants are genuinely inhomogeneous, then apply the contrapositive of a general lemma about homogeneous ideals to certify that the whole ideal cannot be homogeneous. The paper also proposes a \"mutation\" procedure that would give necessary and sufficient conditions for standard homogeneity, while conceding that it has not yet yielded a concrete classification. The sufficient condition thus provides a combinatorial certificate that a Schubert patch is not cut out by homogeneous equations.","feed_headline":"Path test flags inhomogeneous Kazhdan-Lusztig ideals","feed_subtitle":"Permutation-matrix paths and a divisibility check certify that Schubert-patch ideals resist homogeneous generation.","key_machinery":"The load-bearing object is the specialized matrix $Z^{(v)}$, with entries that are $0$, $1$, or an indeterminate, whose southwest submatrices supply the defining minors of $I_{v,w}$. The central mechanism is the path model for determinants: a nonzero path in a minor $A$ chooses one nonzero entry from each column in distinct rows, and Lemma 6.1.1 identifies nonsingularity of $A$ with existence of such a path, since distinct nonzero paths cannot cancel. Theorem 6.1.4 then characterizes a minor having inhomogeneous determinant exactly by the presence of a column of the form $(0,\\dots,0,1,\\dots,z,\\dots)^t$ with a nonzero path through the indeterminate $z$; Lemma 6.1.3 makes that condition checkable by a finite list of inequalities in $v$ and $v^{-1}$. Over this, the decisive algebraic input is Lemma 5.0.1, claiming that in any homogeneous ideal an inhomogeneous generator must have a homogeneous component all of whose monomials are divisible by monomials of other generators or components; the algorithm's final test is precisely the contrapositive of that lemma, with divisibility between monomials of minors decided by Theorems 6.1.1--6.1.3.","core_discovery":"On the paper's own terms, the central discovery is that inhomogeneity of a Kazhdan-Lusztig ideal $I_{v,w}$ can be certified by a purely combinatorial inspection of its defining minors. For a pair $(v,w)$ with $\\tilde R_v \\le \\tilde R_w$ and $w \\neq n\\,n{-}1\\cdots 1$, the procedure reduces the defining minors to an essential set, discards those with singular determinant using the path criteria of Lemmas 6.1.1--6.1.3 and Proposition 6.1.1, and then applies Theorem 6.1.4 to single out minors with inhomogeneous determinant. The final step declares $I_{v,w}$ inhomogeneous whenever the contrapositive of Lemma 5.0.1, a general divisibility condition on homogeneous components, is satisfied; in the KL-ideal setting Remark 6.1.0.3 strengthens the conclusion to divisibility by monomials of other defining minors. Example 8.0.2 runs this algorithm explicitly and produces, for every $n \\ge 6$, a pair $(v,w)$ outside the previously known homogeneous range for which $I_{v,w}$ is inhomogeneous. The authors are explicit that the test is sufficient for inhomogeneity, not necessary.","pith_inferences":["The path criterion in Theorem 6.1.4 resembles a reachability condition in the directed graph of nonzero entries of $Z^{(v)}$, suggesting that the inhomogeneity test could be implemented in polynomial time by a depth-first search rather than by computing determinants.","If Lemma 5.0.1 is repaired or replaced by a correct divisibility lemma, the same contrapositive scheme would convert the algorithm from a sufficient test into a complete one, settling the homogeneity problem for all pairs in a fixed $S_n$.","Because the defining minors are $0/1$/indeterminate matrices, the same path-and-divisibility machinery may apply to other determinantal ideals of Schubert type, not only KL-ideals, whenever their generators have the same column structure.","The mutation procedure is randomized as stated, but its termination condition is a finite system of monomial divisibility equations; restricting to paths would let a computer search decide, for a fixed $n$, whether the procedure terminates for all pairs, giving data toward a conjectural classification by pattern avoidance."],"forward_implications":["For every $n \\ge 6$, Example 8.0.2 yields pairs $(v,w)$ outside the previously known homogeneous range for which the KL-ideal is inhomogeneous, showing that the earlier sufficient conditions are not necessary for homogeneity.","Whenever the algorithm's final divisibility test finds an uncovered homogeneous component, no set of homogeneous polynomials can generate $I_{v,w}$; inhomogeneity is certified in finite time from permutation data alone.","The essential-set reduction means the test only needs the defining minors attached to the essential set, not all southwest submatrices, so the certificate is computable on a reduced generating set.","If the mutation procedure terminates for every homogeneous component of every generator, then Theorem 9.1.1 certifies homogeneity; a successful mutation run would settle standard homogeneity for the pair in question."],"supporting_citations":[{"why":"defines the patch of a Schubert variety at a torus-fixed point, the geometric object whose defining ideal is the KL-ideal.","marker":"[KL79]"},{"why":"supplies the essential-set reduction used to discard redundant defining minors in the algorithm.","marker":"[Ful92]"},{"why":"fixes the rank-matrix and $Z^{(v)}$ conventions for Kazhdan-Lusztig ideals that the paper works with throughout.","marker":"[WY12]"},{"why":"provides the known sufficient conditions for standard homogeneity whose complement the paper's examples occupy.","marker":"[Ney23]"}],"fun_headline_variants":["Path test certifies inhomogeneous Kazhdan-Lusztig ideals","Combinatorial minors check flags inhomogeneous Kazhdan-Lusztig ideals","Sufficient path test reveals new inhomogeneous Kazhdan-Lusztig ideals","Divisibility conditions certify inhomogeneity of Kazhdan-Lusztig ideals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 5.0.1, the claim that in any homogeneous ideal with an inhomogeneous generator, some homogeneous component has every monomial divisible by a monomial from another generator or component; the paper's proof of this lemma is not completed, so if the lemma is false the entire inhomogeneity criterion collapses.","fun_headline_variants_meta":{"raw":{"variants":["Path test certifies inhomogeneous Kazhdan-Lusztig ideals","Combinatorial minors check flags inhomogeneous Kazhdan-Lusztig ideals","Sufficient path test reveals new inhomogeneous Kazhdan-Lusztig ideals","Divisibility conditions certify inhomogeneity of Kazhdan-Lusztig ideals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000938,"raw_usage":{"total_tokens":3955,"prompt_tokens":834,"completion_tokens":3121,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":450,"completion_tokens_details":{"reasoning_tokens":3040}},"tokens_in":450,"tokens_out":3121,"duration_ms":23864,"temperature":1.0,"reasoning_tokens":3040,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:23:33.536128+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The lemma would be refuted by a concrete homogeneous ideal $I=\\langle f_1,\\dots,f_m\\rangle$ over $\\mathbb{C}$ with an inhomogeneous generator $f_i$ such that every homogeneous component $h_{ij}$ has at least one monomial divisible by no monomial of any other generator and by no monomial of any other component of $f_i$; checking the \"cannot contribute\" step in Case 2 of the lemma's proof on a small example would settle the point.","supporting_citations":[],"review_version":2}