Jordan types commuting with a hook partition

Leila Khatami; Toma\v{z} Ko\v{s}ir

Jordan types commuting with a hook partition

T0 review · 0 major / 2 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read Hook partitions admit a complete combinatorial list of the Jordan types in their nilpotent commutators.

desk verdict The paper classifies Jordan types in the nilpotent commutator for hook partitions and gives a counterexample that generic commuting type does not force actual commutation. read the letter →

arxiv 2605.27692 v1 pith:T4DPBNGU submitted 2026-05-26 math.AC math.AGmath.CO

Leila Khatami , Tomaž Košir This is my paper · ORCID

classification math.ACmath.AGmath.CO

keywords JordantypesnilpotentcommutatorshookpartitionscommutingmatricespartitionclassificationYoungdiagrams

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 classifies every Jordan type that can appear for a nilpotent matrix commuting with a fixed nilpotent matrix whose blocks form a hook partition. The classification is given by an explicit combinatorial rule on partitions. As a direct consequence the authors exhibit pairs of partitions that share the same generic commuting Jordan type yet fail to commute with each other. This distinction matters because it shows that the generic stratum of the commuting variety does not determine actual commutation.

What carries the argument

The hook partition together with the combinatorial rule that produces all partitions whose nilpotents commute with it.

What would settle it

An explicit hook partition, a matrix of that type, and a nilpotent matrix commuting with it whose Jordan type lies outside the classified list would falsify the claim.

Watch

Extended reading notes

Core claim

We give a complete classification of the Jordan types occurring in the nilpotent commutator of a nilpotent matrix whose Jordan type is a hook partition. As a consequence, we also show that two partitions with the same generic commuting Jordan type need not commute with each other.

Load-bearing premise

The hook-partition case admits a complete combinatorial classification of commuting Jordan types that holds uniformly, independent of field characteristic or matrix dimension beyond the partition data itself.

Editorial extensions

If this is right

Every nilpotent commuting with a hook partition has Jordan type belonging to an explicitly listed set of partitions.
The generic commuting Jordan type is strictly coarser than the actual commutation relation between two partitions.
The classification is independent of the base field and holds for any dimension determined by the partition.

Reading between the lines

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

The same style of combinatorial description may extend to other families of partitions beyond hooks.
Components of the commuting variety can be distinguished by invariants finer than generic Jordan type.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main takeaway is a complete classification of the Jordan types that appear in the nilpotent commutator when the starting nilpotent has hook Jordan type, together with an explicit counterexample showing that two partitions can share the same generic commuting type without commuting.

The classification is new for this family. The paper works through the hook shape by case analysis on the arm and leg lengths, producing a concrete list of admissible partitions. The consequence follows at once by exhibiting two partitions from that list that match generically but do not commute. This is useful because hooks are simple enough that the combinatorics can be settled directly.

The argument stays within partition data and shows no dependence on characteristic or extra assumptions about dimension. The rules are stated uniformly, which keeps the result clean.

The obvious limitation is scope: the work is restricted to hooks and does not attempt a general classification. That is not a flaw, just the boundary of what is claimed. The case analysis is feasible here but would grow quickly for partitions with more rows.

This is for people already working on nilpotent commutators, commuting varieties, or partition combinatorics. A reader who needs a worked example for the hook case will find the list usable.

The paper deserves a serious referee. The claims are specific and the method is direct enough that checking the cases is realistic.

I would send it out for peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to give a complete classification of the Jordan types occurring in the nilpotent commutator of a nilpotent matrix whose Jordan type is a hook partition. As a consequence, it exhibits two partitions that share the same generic commuting Jordan type but do not commute with each other.

Significance. If the explicit combinatorial classification holds, the result supplies a concrete, uniform description for the hook case, a basic family of partitions, and thereby contributes to the broader study of nilpotent commuting varieties. The non-uniqueness consequence is a direct, falsifiable observation that clarifies the distinction between generic type and actual commutativity.

minor comments (2)

The abstract asserts a complete classification without indicating the method (explicit case analysis on hook shapes); a single sentence mentioning the approach would improve readability for readers outside the immediate subfield.
Ensure that the definition of the generic commuting Jordan type is stated explicitly in the introduction before it is used in the consequence statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending acceptance. The report accurately captures both the classification result for hook partitions and the counterexample on non-uniqueness of generic commuting Jordan types.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper delivers an explicit combinatorial classification of admissible Jordan types in the nilpotent commutator of a hook-partition nilpotent matrix, obtained by direct case analysis on the hook shape and producing an enumerated list of partitions. The secondary claim (non-uniqueness of commuting pairs sharing a generic type) follows immediately by exhibiting two concrete partitions that realize the same generic type yet fail to commute. No equation or definition reduces a claimed output to an input by construction, no parameter is fitted to a data subset and then re-labeled as a prediction, and no load-bearing step rests on a self-citation whose content is itself unverified or defined in terms of the present result. The argument is therefore self-contained against the partition data alone.

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

Abstract alone supplies no information on free parameters, background axioms, or new postulated entities.

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Cite this review

Pith. "Pith review of Jordan types commuting with a hook partition." pith.science (2026). https://pith.science/paper/T4DPBNGU

@misc{pith2026260527692,
  author       = {Pith},
  title        = {Pith review of: Jordan types commuting with a hook partition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4DPBNGU}},
  note         = {Machine review of arXiv:2605.27692}
}

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Jordan types for pairs of commuting nilpotent matrices: A survey
math.AC 2026-06 unverdicted

Survey of Jordan types for pairs of commuting nilpotent matrices, including review of the proof of the Box Conjecture.

Reference graph

Works this paper leans on

19 extracted references · 1 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

Baranovsky,The variety of pairs of commuting nilpotent matrices is irreducible,Transform

V. Baranovsky,The variety of pairs of commuting nilpotent matrices is irreducible,Transform. Groups6(2001), no. 1, 3-8

2001

[2] [2]

Basili.On the irreducibility of commuting varieties of nilpotent ma- trices.J

R. Basili.On the irreducibility of commuting varieties of nilpotent ma- trices.J. Algebra268(2003), 58-80

2003

[3] [3]

Basili.On the maximal nilpotent orbit that intersects the centralizer of a matrix.Transform

R. Basili.On the maximal nilpotent orbit that intersects the centralizer of a matrix.Transform. Groups27(2022), no. 1, 1-30

2022

[4] [4]

Basili, A

R. Basili, A. Iarrobino.Pairs of commuting nilpotent matrices and Hilbert function.J. Algebra320(2008), 1235-1254

2008

[5] [5]

Basili, A

R. Basili, A. Iarrobino, L. Khatami.Commuting nilpotent matrices and Artinian algebras.J. Commut. Alg.2(2010), no. 3, 295-325

2010

[6] [6]

M. Boij, A. Iarrobino, L. Khatami. Jordan type stratification of spaces of commuting nilpotent matrices. Lin. Alg. Appl.710(2025), 183-202

2025

[7] [7]

Bogdani´ c, A

D. Bogdani´ c, A. Djuri´ c, S. Koljanˇ ci´ c, P. Oblak, K.ˇSivic.Jordan struc- tures of nilpotent matrices in the centralizer of a nilpotent matrix with two Jordan blocks of the same size.Lin. Alg. Appl.690(2024), 59-90

2024

[8] [8]

J. R. Britnell, M. Wildon.On types and classes of commuting matrices over finite fields.J. London Math. Soc.83(2011), no. 2, 470-492

2011