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On the homogeneity problem of the Kazhdan-Lusztig ideals

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A serious but currently unproven inhomogeneity test for KL ideals; the path criterion and the explicit examples are worth a referee's time. read the letter →

arxiv 2506.20153 v2 pith:KWGSWSPF submitted 2025-06-25 math.CO

Adhip Ganguly , Shyamashree Upadhyay This is my paper · ORCID
classification math.CO MSC 05E1405E4014N1013C70
keywords Kazhdan-LusztigidealstandardhomogeneousinhomogeneousSchubertvarietydeterminantalpathessentialsetrankmatrix
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

4 major / 6 minor

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.

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 (4)
  1. [Section 5, Lemma 5.0.1, Case 2] 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.
  2. [Section 5, Lemma 5.0.1, Case 4(a)] 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.
  3. [Remark 4.2.0.2] 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}.
  4. [Example 8.0.2] 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'.
minor comments (6)
  1. [Section 1, page 2] 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.
  2. [Section 6.1, Observation 2 proof] 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.
  3. [Lemma 6.1.2] 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.
  4. [Example 8.0.1] 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.
  5. [Section 7.1] 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.
  6. [Section 9] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inhomogeneity test applies a general algebraic lemma to independently derived combinatorial criteria; no claim reduces to its own inputs.

full rationale

The derivation chain is not circular. The central claim is that the §7.1 procedure gives sufficient conditions for a Kazhdan-Lusztig ideal I_{v,w} to be inhomogeneous. The procedure reduces to independent mathematical inputs: Lemma 5.0.1 is a general algebraic statement about arbitrary ideals in C[x], not a statement about KL ideals; Theorem 6.1.4 and the path lemmas characterize when a single defining minor has inhomogeneous determinant; Remark 4.2.0.2 is an external Fulton-style simplification of the generating set; Lemma 3.1.1 and Theorem 3.1.1 discard trivial cases. The final sentence in §7.1, 'We declare the KL-ideal I_{v,w} to be inhomogeneous if it satisfies the sufficient condition(s) given by the contrapositive statement of Lemma 5.0.1', is a logical application of that lemma, not a redefinition: inhomogeneity was already defined independently in §1 as the failure of standard homogeneity. No parameter is fitted to data, no claim is justified solely by a self-citation (the cited works [Ney23], [WY12], [WY08], [KL79], [Ful92] are by other authors and are used as benchmarks or external ingredients), and no ansatz is imported from the present authors' prior work. The manuscript itself discloses unfinished aspects: §9 admits that 'we have not yet been able to deduce any concrete set of necessary as well as sufficient conditions' from mutation, and Remark 4.2.0.2 says 'One can prove a lemma similar to lemma 3.10 in [Ful92]' without giving the proof. These are rigor and completeness concerns about the validity of Lemma 5.0.1's proof and the essential-set reduction, not circular dependence of the conclusion on itself. No circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no adjustable parameters or new objects; its assumptions come from prior literature and from Lemma 5.0.1, whose proof is incomplete. The main algebraic weight is carried by unproven Lemma 5.0.1 and the unproven essential-set reduction.

assumptions (4)
  • domain assumption Kazhdan-Lusztig ideal is generated by the defining minors given in Section 2.1, following definitions from [KL79] and [WY23].
    The paper takes this as the definition of I_{v,w}; no proof that these minors generate is supplied beyond the sketch in Section 1.
  • domain assumption Essential set reduction (Remark 4.2.0.2): it suffices to take minors corresponding to the essential set E defined in Section 4.2.
    Asserted as 'one can prove a lemma similar to lemma 3.10 in [Ful92]' but no proof is given; the algorithm in Section 7 depends on it.
  • ad hoc to paper Lemma 5.0.1: for a homogeneous ideal generated by possibly inhomogeneous polynomials, some homogeneous component of an inhomogeneous generator has every monomial divisible by a monomial of another generator or another component.
    This is the load-bearing algebraic premise for the contrapositive used in Section 7.1; the proof in Section 5 has gaps and the statement is used as an unproved general principle.
  • domain assumption The equivalence between the rank-matrix definition of Schubert varieties and the orbit-closure definition (stated in Section 1 as 'One can show that these two definitions are equivalent').
    Standard from [Ful92] and [WY23], used implicitly throughout the paper.

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Pith. "Pith review of On the homogeneity problem of the Kazhdan-Lusztig ideals." pith.science (2026). https://pith.science/paper/KWGSWSPF

@misc{pith2026250620153,
  author       = {Pith},
  title        = {Pith review of: On the homogeneity problem of the Kazhdan-Lusztig ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KWGSWSPF}},
  note         = {Machine review of arXiv:2506.20153}
}
read the original abstract

In this paper, we identify some sufficient conditions for a Kazhdan-Lusztig ideal to be inhomogeneous. Also, we attempt to approach the problem of giving some necessary and sufficient conditions for a Kazhdan-Lusztig ideal to be "standard homogeneous".

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Works this paper leans on

6 extracted references · 5 canonical work pages

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    Representations of coxeter groups and hecke algebras

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    A gr \"o bner basis for schubert patch ideals

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    Governing singularities of schubert varieties

    Alexander Woo and Alexander Yong. Governing singularities of schubert varieties. Journal of Algebra , 320(2):495--520, 2008

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    A gr \"o bner basis for kazhdan-lusztig ideals

    Alexander Woo and Alexander Yong. A gr \"o bner basis for kazhdan-lusztig ideals. American Journal of Mathematics , 134(4):1089--1137, 2012

    work page 2012

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    Schubert geometry and combinatorics

    Alexander Woo and Alexander Yong. Schubert geometry and combinatorics. arXiv preprint arXiv:2303.01436 , 2023

    arXiv 2023

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