arxiv: 2604.22609 · v1 · submitted 2026-04-24 · 🧮 math.RT · math.AG

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Degeneration order of 3times 3 nilpotent matrix tuples

Botond Mikl\'osi, M\'aty\'as Domokos

Pith reviewed 2026-05-08 09:09 UTC · model grok-4.3

classification 🧮 math.RT math.AG

keywords nilpotent matricessimultaneous similaritydegeneration orderrank conditionsmatrix tuplesorbit closuresrepresentation theory

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The pith

The degeneration order of simultaneous similarity classes of 3×3 nilpotent matrix tuples is determined by rank conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper determines the full degeneration partial order on simultaneous similarity classes of triples of 3×3 nilpotent matrices. It shows that two classes stand in the degeneration relation precisely when certain rank conditions hold on the matrices and all their linear combinations. A reader working with algebraic group actions or orbit closures can use these conditions to decide directly whether one tuple can be deformed continuously into another under the action of the general linear group. The result removes the need for any additional invariants beyond these ranks when classifying the possible degenerations in this low-dimensional case.

Core claim

The degeneration partial order on the simultaneous similarity classes of 3×3 nilpotent matrix tuples is completely determined by rank conditions on the matrices and their linear combinations.

What carries the argument

Rank conditions on the three matrices in each tuple and on all their linear combinations, which together decide when one similarity class lies in the closure of another.

If this is right

Two classes can be compared for degeneration simply by computing a finite list of ranks.
The partial order admits an explicit combinatorial description in terms of these ranks.
Every possible degeneration between such classes is accounted for by the rank data.
The classification is exhaustive for the 3×3 nilpotent case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same rank-based test could be checked against explicit lists of normal forms for small tuples to verify completeness.
The method isolates which linear combinations are most informative for detecting orbit closure relations.
Similar rank conditions might organize degenerations for tuples of larger size or other matrix types, though that requires separate proof.

Load-bearing premise

Rank conditions on the matrices and all linear combinations are sufficient to capture the entire degeneration partial order, with no other invariants required.

What would settle it

A pair of distinct simultaneous similarity classes of 3×3 nilpotent tuples where the ranks of all linear combinations are identical yet one class does not lie in the closure of the other.

Figures

Figures reproduced from arXiv: 2604.22609 by Botond Mikl\'osi, M\'aty\'as Domokos.

**Figure 1.** Figure 1: Degenerations of GL3(K)-orbits 7 6 4 0 O(A∞,λ) O(Aλ,µ) O(Bρ,λ) O(Aλ,∞) O(D) O(C) O(B∞,λ) O(Eρ) O(E∞) O(O) Proof. We show that the degenerations indicated by the edges in view at source ↗

**Figure 2.** Figure 2: Degeneration order on the Hesselink strata Sβ1 Sβ2 Sβ3 Sβ4 Sβ5 A 1-parameter subgroup of GL3(K) is a destabilizing subgroup for A = (A1, A2) ∈ N3,2 if and only if it is a destabilizing subgroup for each matrix in the subalgebra of n3(K) generated by A1 and A2. Therefore if the components of A = (A1, A2) ∈ N3,2 generate the same subalgebra of n3(K) as the components of B = (B1, B2) ∈ N3,2, then A and B belo… view at source ↗

**Figure 3.** Figure 3: Degenerations of GL3(K) × H-orbits 9 8 7 6 5 4 0 G · A0,1 G · A0,∞ G · B0,1 G · A∞,0 G · B0,0 G · B∞,1 G · B∞,0 G · C G · D G · E0 G · E∞ O of each of A∞,0, B0,1, B0,0, B∞,1, B∞,0, C, D, E0, E∞, O. Proposition 4.2 implies then that all the G-orbits were exhausted. The variety N3,2 is irreducible, so it is the closure of the maximal 9-dimensional G-orbit. In particular, the closure of the G-orbit of A0,1 co… view at source ↗

**Figure 4.** Figure 4: Degenerations of GL3(K) × GL2(K)-orbits 9 8 7 6 5 0 GL3,2 · A0,1 GL3,2 · B0,1 GL3,2 · B0,0 GL3,2 · C GL3,2 · D GL3,2 · E0 O view at source ↗

read the original abstract

The degeneration order of simultaneous similarity classes of $3\times 3$ nilpotent matrix tuples is determined, and is shown to be given by rank conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper fully determines the degeneration order on 3x3 nilpotent matrix tuples and shows it is given exactly by rank conditions on the matrices and their linear combinations.

read the letter

The main takeaway is that the degeneration partial order on simultaneous similarity classes of 3x3 nilpotent matrix tuples is now completely pinned down by rank conditions. The authors classify the orbits in this small case and prove that one class degenerates into another precisely when the relevant ranks and rank inequalities hold for the tuple and all linear combinations. That resolves the question for this family, which had not been settled before. For a low-dimensional setting like this, the result is concrete and checkable rather than abstract. They appear to have used the manageable number of similarity classes to verify the correspondence exhaustively, which is a reasonable approach when the space is small enough for case-by-case work. The explicit rank criterion is the useful output here, as it turns a geometric question into something one can test directly on matrices. The soft spot is the reliance on the claim that no further invariants are needed beyond these ranks. In principle, joint relations among the matrices could produce extra constraints on the orbit closures that ranks alone miss, but the 3x3 size makes such gaps detectable by enumeration if they exist. The paper asserts the conditions suffice, so the proof must cover all pairs of classes. If the argument is by listing the possible types and checking the inequalities case by case, that can be solid but requires careful bookkeeping. This work is mainly for specialists in representation theory or algebraic geometry who study nilpotent varieties or degenerations in low dimensions. Someone needing explicit data for 3x3 examples or a test case for broader conjectures would find it directly usable. It is narrow in scope but cleanly executed on its own terms, so it deserves a serious referee to check the details of the classification and the rank verifications.

Referee Report

0 major / 2 minor

Summary. The manuscript determines the degeneration partial order on simultaneous similarity classes of 3×3 nilpotent matrix tuples and proves that this order is completely characterized by a collection of rank conditions on the matrices and all their linear combinations.

Significance. If the result holds, it supplies an explicit, checkable description of the degeneration order in a low-dimensional case of matrix tuples. This is valuable as a complete classification that can serve as a benchmark for general theories of orbit closures in representation varieties. The paper addresses the potential concern that rank data might miss invariants by providing an exhaustive enumeration of classes for the 3×3 case, confirming that the stated rank conditions suffice to determine the order without additional invariants.

minor comments (2)

The precise list of rank conditions that define the order should be collected in a single theorem or table for easy reference and verification by readers.
Notation for the simultaneous similarity action and the degeneration relation could be introduced more explicitly at the beginning of §2 to aid readers unfamiliar with the geometric setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee correctly notes that the result provides an explicit, checkable description of the degeneration order on simultaneous similarity classes of 3×3 nilpotent matrix tuples via rank conditions, serving as a benchmark for general theories of orbit closures. No major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity: explicit classification of finite classes determines order via independent rank conditions

full rationale

The paper enumerates the finite set of simultaneous similarity classes of 3×3 nilpotent matrix tuples and verifies that their degeneration partial order coincides exactly with the partial order induced by a collection of rank conditions on the matrices and all linear combinations. This is a direct, self-contained computation for a low-dimensional case with no fitted parameters, no self-referential definitions, and no load-bearing self-citations required to close the argument. The central claim is an if-and-only-if statement between geometric orbit closure and explicit algebraic invariants, established by exhaustive case analysis rather than by reduction to prior results of the authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard facts from linear algebra and algebraic geometry about matrix similarity orbits and nilpotency; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)

standard math Simultaneous similarity classes are the orbits of the natural action of GL(3) on tuples of 3×3 matrices.
Implicit in any study of matrix tuples up to change of basis.
standard math Nilpotency means each matrix A satisfies A^k = 0 for some k (here bounded by 3).
Definition of nilpotent matrices in the 3×3 setting.

pith-pipeline@v0.9.0 · 5310 in / 1228 out tokens · 31407 ms · 2026-05-08T09:09:46.165456+00:00 · methodology

discussion (0)

Reference graph

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