Identifying Partitions with maximum commuting orbit $Q=(u,u-r)$

Anthony Iarrobino; Leila Khatami; Mats Boij

REVIEW 3 major objections 4 minor 12 references

Identifying Partitions with maximum commuting orbit $Q=(u,u-r)$

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For Q=(u,u-r), the geometric partition table and the Burge-code table coincide at every index (k,l).

desk verdict A clean proof of a deferred table identification whose load-bearing lemmas are cited from an unpublished preprint; conditional but worth refereeing. read the letter →

arxiv 2411.18340 v1 pith:KNTQS35Y submitted 2024-11-27 math.AC math.AGmath.COmath.RA

Mats Boij , Anthony Iarrobino , Leila Khatami This is my paper · ORCID

classification math.ACmath.AGmath.COmath.RA MSC 15A2705A1713E1014A0515A20

keywords JordantypecommutingnilpotentmatricescommutatorBurgecorrespondencestablepartitionordermatrixtablelocusequations

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 proves that two independent ways of listing the same set of partitions agree entry by entry. For a stable two-part partition $Q=(u,u-r)$, the geometric table $\mathcal T(Q)$ of [IKVZ2] records the Jordan types whose maximal commuting nilpotent orbit is $Q$, while the Burge-correspondence table of [BIK] records the same types by codes. The paper shows that the entry $P_{k,l}(Q)$ in the geometric table is exactly the entry $P^Q_{k,l}$ with Burge code $\alpha^{u-r-l}\beta^l\alpha^{r-k}\beta^k\alpha$, and that the equations $E^Q_{k,l}$ define the locus of each such type. This settles a 2015 conjecture and makes the stratification of the nilpotent commutator for two-part $Q$ explicit.

What carries the argument

The central machinery is the order matrix $T$, whose entries are the orders of the four blocks of a matrix $A$ in the nilpotent commutator of $J_Q$: the upper-left block has order $k$, the lower-left block order $r$, and the lower-right block order $l$, with $l' = l \oplus (r-k)$ controlling the effective index. From $T$, the cited lemmas of [BIK] give an explicit formula for the corank of $A^s$; the proof then compares four lines $L_1(s)=(k+l)s$, $L_2(s)=ks+u-r$, $L_3(s)=l's+u-2l'$, and $L_4(s)=u+u-r$, and shows that whichever line is lowest at each $s$ reconstructs the Jordan type $P_{k,l}(Q)$ forced by the equations $E^Q_{k,l}$. The Burge code $\alpha^{u-r-l}\beta^l\alpha^{r-k}\beta^k\alpha$ is the combinatorial counterpart indexing the same partition.

What would settle it

Take $Q=(8,4)$, choose any $(k,l)$ with $1\le k\le 3$ and $1\le l\le 4$, and sample a matrix $A$ whose entries satisfy $E^Q_{k,l}$ with all other entries generic over a finite field; if the Jordan type of $A$ ever differs from $P_{k,l}(Q)$ as listed in Theorem 2.2 - for instance, if the number of parts is not $k+l$ - then Theorem 4.1 is false.

Watch

Extended reading notes

Core claim

The paper establishes that the partition $P_{k,l}(Q)$ appearing in the table $\mathcal T(Q)$ of [IKVZ2] - the table of Jordan types whose maximal commuting nilpotent orbit is the stable two-part partition $Q=(u,u-r)$ - is the same partition as $P^Q_{k,l}$ defined in [BIK] by the Burge code $\alpha^{u-r-l}\beta^l\alpha^{r-k}\beta^k\alpha$. The proof shows that the equations $E^Q_{k,l}$, conjectured in [IKVZ1] to cut out the locus of $P_{k,l}(Q)$, force a matrix $A\in \mathcal N(Q)$ with generic remaining entries to have Jordan type $P_{k,l}(Q)$. Consequently the geometric A/B/C-type table and the Burge-code table coincide at every index $(k,l)$ (Corollary 4.2), and the closure of each stratum is defined by $E^Q_{k,l}$ (Corollary 4.3).

Load-bearing premise

The proof depends on two lemmas from the authors' earlier preprint that calculate the rank drops of every power of $A$ from a small 2-by-2 matrix of leading powers; if those lemmas are wrong or carry hidden conditions, the identification fails.

Editorial extensions

If this is right

For every index $(k,l)$ with $1\le k\le r-1$ and $1\le l\le u-r$, the locus of matrices in $\mathcal N(Q)$ having Jordan type $P_{k,l}(Q)$ is defined by the equations $E^Q_{k,l}$, confirming the 2015 conjecture for two-part stable $Q$.
The geometric table $\mathcal T(Q)$ and the Burge-code table $P^Q_{k,l}$ are the same list of $(r-1)(u-r)$ partitions, each with $k+l$ parts, so the A/B/C types and the Burge-code types are the same objects.
For a generic matrix $A$ satisfying $E^Q_{k,l}$, the corank sequence of $A^s$ is governed by the four lines $L_1,\dots,L_4$, so the Jordan type can be read off from the intersection points of those lines.
The closure of the locus of matrices with Jordan type $P_{k,l}(Q)$ is the affine variety $V(E^Q_{k,l})$, giving an explicit defining-equation stratification of $\mathcal N(Q)$ for $Q=(u,u-r)$.

Reading between the lines

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

A natural extension would be to test whether the same order-matrix mechanism identifies a Burge-code table with a geometric table for stable partitions $Q$ with more than two parts; the two-part case suggests the corank formula, not the A/B/C type geometry, is the essential ingredient.
The equality of the two tables makes the stratification for two-part $Q$ effectively combinatorial: the dominance order on the partitions $P_{k,l}(Q)$ should be readable from the Burge codes alone, so closure containments between strata should follow from a word-ordering rule.
Because Theorem 4.1 requires the entries outside $E^Q_{k,l}$ to be generic, the equations alone may cut out a larger variety containing the stratum; solving $E^Q_{k,l}$ at non-generic points would show whether that genericity hypothesis is necessary.

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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

Short story: this is a proof of [BIK, Proposition 3.9], which was announced without proof. The authors show that the partition P_{k,l}(Q) from the IKVZ2 table and the Burge-code partition P^Q_{k,l} from their earlier preprint are the same, by proving that the equations E^Q_{k,l} define the same generic Jordan type. Theorem 4.1 is an explicit, readable case analysis on four linear bounds, and it looks correct as far as I followed it. The Corollary 4.2 identification then falls out, provided you accept the [BIK] setup.

What is genuinely good: the proof technique is straightforward and the paper connects two existing tables in a way that was previously left undone. The case splits are thorough, and the authors are honest that the proof was deferred from [BIK]. The note is short, to the point, and likely useful to the commuting nilpotent matrices community.

Soft spots. The main one is the one the reader flagged: the corank formula in Theorem 4.1, including the tensor-power computation, is imported verbatim from [BIK, Lemmas 3.5 and 3.6], a same-author preprint. That is not a fatal flaw—the paper is explicitly a companion to [BIK]—but the claim that the note is self-contained does not hold. The decisive lemmas are not proved here, and the proof is exactly as secure as they are. The stress-test worry about hidden hypotheses at the boundary k+l = r or r+1 is speculative; I saw no sign of failure. Still, a referee should ask for the lemmas to be stated and either proved or clearly located in the final [BIK]. Also, there is a typo 'L!' for L1 in Case A and a few other small proofreading issues; nothing that obscures the argument.

For whom? This is for specialists in Jordan types of commuting nilpotent matrices. It is not a broad result, but it closes a known gap. I would send it to a referee with normal standards; the main requested revision should be that the [BIK] lemmas be made available or at least restated so the proof is checkable without a second preprint. It is a legitimate, focused contribution.

Recommendation: accept for peer review, with a request for better self-containedness.

Referee Report

3 major / 4 minor

Summary. The note proves that for a stable two-part partition Q=(u,u-r), the partition P_{k,l}(Q) appearing in the IKVZ2 table T(Q) of partitions with maximal nilpotent commutator Q coincides with the partition P^Q_{k,l} defined in [BIK] by a Burge code. The proof (Theorem 4.1) assumes the equations E^Q_{k,l} hold and the remaining entries are generic, computes the corank sequence of powers of A from [BIK, Lemmas 3.5 and 3.6], reconstructs the Jordan type from that sequence, and checks the result against the Table Theorem 2.2. Corollary 4.2 then asserts the equality of the two tables entry by entry, and Corollary 4.3 uses this to identify the locus of each P_{k,l}(Q) with the variety defined by E^Q_{k,l}.

Significance. If correct, the result resolves the identification deferred in [BIK, Proposition 3.9] and confirms for two-part stable Q the equation-locus conjecture from [IKVZ1, Conjecture 4.19]. The proof is a genuine derivation rather than a circular comparison: the Jordan type is obtained from the corank sequence and then matched with the pre-existing table. The detailed four-line case analysis in Section 4 is a useful, explicit verification. The main limitation is that the entire rank computation is imported from two lemmas in the same authors' unpublished preprint [BIK], so the central equality is only as secure as those lemmas.

major comments (3)

[Section 4, Theorem 4.1] The proof does not prove [BIK, Lemma 3.6] or [BIK, Lemma 3.5]; these lemmas supply the full corank formula, including the tensor-power expression (T^{⊗s})_{11}. Since every subsequent step in the case analysis and the final identification with the table depend on that formula, the equality in Corollary 4.2 is not established within this note. Please state the two lemmas completely and either include proofs (an appendix would be acceptable) or provide a precise reference to a version that can be checked; in addition, verify the boundary cases k+l=r, k+l=r+1, and k=l'=r/2, where divisions by k-l' or l' appear in the intersection formulas.
[Section 4, Cases A and B] The definitions of the two cases are garbled as printed: Case A reads 'L3(s) ≥ L!(s) ⊕ L2(s) ⊕ L4(s)' and Case B reads 'L2(s) ≥ L!(s) ⊕ L2(s) ⊕ L4(s)', where 'L!' should presumably be 'L1'. With the literal text, the Case B inequality is automatically true because L2(s) appears on both sides, so the case split does not define a partition of the possibilities. This needs correction because the proof relies on the claim that Cases A, B, and C cover all possible orderings of the four lines.
[Section 4, Theorem 4.1] The hypothesis 'and that the rest of the entries of A are generic' is never formalized. The proof uses the corank sequence and the recovery of the Jordan type from it, but the statement should specify the precise open condition on A (or cite the exact [BIK] result showing that the corank formula holds for every matrix in the locus defined by E^Q_{k,l}, so that no genericity assumption is needed). Without this, the theorem does not clearly identify the locus of P_{k,l}(Q) rather than merely some dense subset.

minor comments (4)

[Section 3, Definition 3.1] The symbol ⊕ is used throughout for what appears to be the minimum (for example l′ = l ⊕ (r−k), and later corank A^s = (k+l)s ⊕ ((T^{⊗s})_{11}+u−r) ⊕ (u+u−r)), but it is never defined in this note. Please define it explicitly.
[Theorem 2.2] The statement of the Table Theorem uses the symbols k_t, c_t, d_t, and q_t without defining them. Since the introduction says the note is meant to be self-contained, these definitions should be included, or the statement should refer precisely to the version in [IKVZ2].
[Section 4, proof of Theorem 4.1] The notions Utop, Ubottom, and Umiddle (the 'U-chains') are used in the verification that Q(P)=(u,u−r) but are not defined in the note. Please give a definition or a precise reference to [IKVZ2].
[Corollary 4.3] The proof says 'This follows from Corollary 4', but the intended reference is Corollary 4.2; please correct the cross-reference.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity: the table identity is derived from corank data, but the proof leans on unproved same-author lemmas from [BIK].

full rationale

The paper's derivation chain is: Section 3 recalls the auxiliary matrix M_B and defines the equations E^Q_{k,l}; Theorem 4.1 assumes E^Q_{k,l} and generic entries, imports from [BIK] the corank formula 'corank A^s = (k+l)s ⊕ ((T^{⊗s})_{11} + u − r) ⊕ (u + u − r)' (Theorem 4.1 proof, 'By [BIK, Lemma 3.6]') and the tensor-power values ('Moreover, by [BIK, Lemma 3.5]'), computes the corank sequence, reconstructs the Jordan type, and matches it with the type A/B/C formulas of Theorem 2.2 from [IKVZ2]. Corollary 4.2 then concludes the identity with P^Q_{k,l} because [BIK] had shown the same equations define its locus. No step defines P^Q_{k,l} in terms of P_{k,l}(Q), or vice versa: P^Q_{k,l} is defined by its Burge code (Equation 3), and P_{k,l}(Q) by the [IKVZ2] table. The equality is therefore the output of the argument, not an input. The only caveat is that the proof leans on [BIK, Lemmas 3.5 and 3.6] and on [BIK]'s equation-locus identification, all from the same authors' preprint and not proved or independently verified here; the Introduction explicitly says 'this is [BIK, Proposition 3.9], whose proof we deferred to here.' That is a load-bearing self-citation and a missing-support risk, but it is not circularity because the cited lemmas do not assume the target equality. If the lemmas are wrong, the conclusion is unsupported, but it is not assumed. Hence score 2: no definitional circularity, only a self-citation burden.

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

The argument is pure mathematics over arbitrary u and r with no data fitting. The only non-trivial external inputs are the two corank lemmas from the same-authors' preprint [BIK], plus the prior table theorem from [IKVZ2]. These are cited rather than proved in this note.

assumptions (4)

domain assumption Validity of [BIK, Lemma 3.6]: corank A^s follows from the order matrix T by the formula (k+l)s ⊕ (sk+u-r) ⊕ (u+u-r) in the applicable cases.
Invoked at the start of the proof of Theorem 4.1; not proved in this note, and [BIK] is an unpublished preprint by the same authors.
domain assumption Validity of [BIK, Lemma 3.5]: the parity-dependent formula for (T^{⊗s})_{11} in terms of k, r, and l'.
Same provenance as the corank lemma; used throughout the line comparisons in Cases A, B, and C of Theorem 4.1.
domain assumption The Table Theorem 2.2 and 2.3 of [IKVZ2] correctly enumerates D^{-1}(Q) with the types A, B, C and the indexing (k,l).
Defines the object P_{k,l}(Q) being compared; cited from prior published work rather than proved here.
standard math The corank sequence of a nilpotent matrix uniquely determines its Jordan partition.
Used in the 'Equation to partition' section to reconstruct P from corank A^s; a standard fact in nilpotent matrix theory.

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Pith. "Pith review of Identifying Partitions with maximum commuting orbit $Q=(u,u-r)$." pith.science (2026). https://pith.science/paper/KNTQS35Y

@misc{pith2026241118340,
  author       = {Pith},
  title        = {Pith review of: Identifying Partitions with maximum commuting orbit $Q=(u,u-r)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KNTQS35Y}},
  note         = {Machine review of arXiv:2411.18340}
}

abstract

The authors here show that the partition $P_{k,l}(Q)$ in the table $\mathcal T(Q)$ of partitions having maximal nilpotent commutator a given stable partition $Q$, defined in [IKVZ2], is identical to the analogous partition $P_{k,l}^Q$ defined by the authors in [BIK] using the Burge correspondence.

Figures

Figures reproduced from arXiv: 2411.18340 by the authors.

**Figure 1.** Comparing terms in the expression for corank As in Case A of the proof of Theorem 4.1. In all cases above, we can write P = ([u] k , [u − r] l ). Looking at U-chains in P, we have |Utop| = u |Ubottom| = u − r + 2k. We claim that u − r + 2k ≤ u. In fact, if u − r + 2k > u then 2k > r. By definition we also know that r ≥ k + l ′ . Thus the inequality 2k > r implies that k > l′ . However, if this is the case, then for … view at source ↗

**Figure 2.** Comparing terms in the expression for corank As in Case B of the proof of Theorem 4.1. P =    (s4 + 3)e4 , (s4 + 2)l ′−e4 , (s3 + 1)e3 , (s3) k+l−l ′−e3 , if e3, e4 > 0 (s4 + 3)e4 , (s4 + 2)l ′−e4 , (s3) k+l−l ′ , if e4 > 0 and e3 = 0 (s4 + 2)l ′ , (s3 + 1)e3 , (s3) k+l−l ′−e3 , if e4 = 0 and e3 > 0 (s4 + 2)l ′ , (s3) k+l−l ′ , if e3 = e4 = 0. In all cases above, we can wri… view at source ↗

**Figure 3.** Comparing terms in the expression for corank As in Case C of the proof of Theorem 4.1. We also have x3,4 = u−r l + 2 = x1,2. Thus if we let s1 = ⌊ u−r l ⌋, then we have ⌊x2,3⌋ = s1 + 1, and corank As = s(k + l), for s = 1, . . . , s1 corank As1+1 = (s1 + 1)k + u − r, corank As1+2 = s1l + u, corank As = u + u − r, for s ≥ s1 + 2. See figure 3 for a visualization of the linear terms affecting corank As in this case. T… view at source ↗

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Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Basili, On the irreducibility of commuting varieties of nilpotent matrices, J

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work page 2003

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work page 2022

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work page 2024

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work page 1981

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