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arxiv: 2502.10571 · v2 · pith:GMCVMNE4new · submitted 2025-02-14 · 🧮 math.RA · math.SP

Perron matrix semigroups

Vladimir Yu. Protasov This is my paper

Pith reviewed 2026-05-23 02:48 UTC · model grok-4.3

classification 🧮 math.RA math.SP
keywords Perron semigroupinvariant conematrix semigroupirreducible semigroupspectral radiusconvex conemultiplicative semigroupjoint invariance
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The pith

Every irreducible Perron semigroup of matrices shares a common invariant cone under mild assumptions.

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

The paper studies multiplicative semigroups of real d by d matrices in which every matrix has an eigenvalue equal to its spectral radius. Such semigroups are called Perron. It is already known that if the matrices all leave some proper convex cone invariant, then the semigroup is Perron. The central result establishes the converse direction: under irreducibility plus some mild extra conditions, a Perron semigroup must admit a single common invariant cone for all its matrices. The work also classifies the exceptional Perron semigroups that lack any common invariant cone, completely for dimension at most four and partially for larger dimensions.

Core claim

Every irreducible Perron semigroup possesses a common invariant cone, provided that some mild assumptions are satisfied. This supplies concrete conditions under which a collection of matrices is guaranteed to share an invariant cone.

What carries the argument

A Perron semigroup: a multiplicative semigroup of matrices in which each matrix possesses a Perron eigenvalue (an eigenvalue equal to the spectral radius). Irreducibility of the semigroup is the additional structural assumption that converts the Perron property into the existence of a joint invariant cone.

If this is right

  • Any irreducible set of matrices whose generated semigroup is Perron will admit a joint proper convex cone that is left invariant by every matrix in the set.
  • The result supplies a sufficient condition, phrased in terms of the spectral-radius eigenvalue property, for a family of matrices to be simultaneously positive with respect to some cone.
  • For dimensions two, three and four every Perron semigroup is either cone-preserving or belongs to one of the explicitly listed exceptional classes.
  • In dimensions greater than four there exist infinite families of irreducible Perron semigroups that fail to possess any common invariant cone.

Where Pith is reading between the lines

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

  • The classification in low dimensions may allow algorithmic checks for the existence of a common cone by testing membership in the exceptional lists.
  • The mild assumptions are likely to involve boundedness or compactness properties of the semigroup; relaxing them could produce further counter-examples.
  • The same cone-invariance question arises for semigroups acting on infinite-dimensional spaces or on ordered Banach spaces, where the Perron property can be formulated via the spectral radius.

Load-bearing premise

The semigroup must be irreducible and must satisfy the unspecified mild assumptions required for the converse statement to hold.

What would settle it

An explicit irreducible Perron semigroup, obeying the mild assumptions, that nevertheless possesses no common invariant cone for all its matrices.

read the original abstract

We consider multiplicative semigroups of real dxd matrices. A semigroup S is called Perron if each of its matrices has a Perron eigenvalue, i.e., an eigenvalue equal to the spectral radius. If all matrices of S leave a proper convex cone invariant, then S is Perron. Our main result asserts the converse: every irreducible Perron semigroup possesses a common invariant cone, provided that some mild assumptions are satisfied. This gives conditions for a set of matrices to share a common invariant cone, which is an important property widely studied in the literature. Then we address the problem to characterize the exceptions, when a Perron semigroup does not have an invariant cone. For d\le 4, all Perron semigroups are classified. For higher dimensions~$d$, several classes of such semigroups are found.

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.

Referee Report

1 major / 0 minor

Summary. The paper studies multiplicative semigroups of real d×d matrices. A semigroup S is Perron if every matrix has a Perron eigenvalue (an eigenvalue equal to the spectral radius). The manuscript proves that if all matrices leave a proper convex cone invariant then S is Perron, and asserts the converse: every irreducible Perron semigroup possesses a common invariant cone provided mild assumptions hold. It classifies all Perron semigroups for d≤4 and exhibits several classes of exceptions in higher dimensions.

Significance. If the main result holds, it supplies verifiable conditions under which a collection of matrices shares a common invariant cone, a property of independent interest in the literature on positive systems and joint spectral radius. The explicit classification for d≤4 and the concrete families constructed for d>4 constitute falsifiable, checkable output that strengthens the overall program.

major comments (1)
  1. Abstract and §1: the central converse claim is stated only under unspecified 'mild assumptions' and without a definition of irreducibility. These omissions are load-bearing because the result is asserted to hold precisely when the assumptions are satisfied; without their explicit statement the claim cannot be verified or tested.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and for highlighting the need for greater precision in the presentation of the main result. We address the comment below.

read point-by-point responses

  1. Referee: [—] Abstract and §1: the central converse claim is stated only under unspecified 'mild assumptions' and without a definition of irreducibility. These omissions are load-bearing because the result is asserted to hold precisely when the assumptions are satisfied; without their explicit statement the claim cannot be verified or tested.

    Authors: We agree that the abstract and §1 must state the mild assumptions explicitly and define irreducibility at the outset, as these conditions determine the precise scope of the converse. In the revised manuscript we will insert the full statement of the assumptions (as they appear in the body) into the abstract and the opening paragraphs of §1, together with the definition of irreducibility, so that the claim is self-contained and verifiable from the first page. revision: yes

Circularity Check

0 steps flagged

No circularity: direct proof of converse theorem

full rationale

The paper states a mathematical result: every irreducible Perron semigroup has a common invariant cone under mild assumptions, as the converse to the known implication that invariant cones imply the Perron property. The provided abstract and description contain no equations, fitted parameters, self-citations used as load-bearing uniqueness theorems, or ansatzes smuggled via prior work. The classification for d ≤ 4 is presented as explicit enumeration rather than a renamed empirical pattern. No step reduces by construction to its own inputs; the derivation is self-contained as a standard theorem proof in matrix semigroup theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or non-standard axioms are visible.

axioms (1)
  • standard math Standard properties of eigenvalues, spectral radius, and convex cones in finite-dimensional real vector spaces.
    Invoked in the definition of Perron eigenvalue and invariant cone.

pith-pipeline@v0.9.0 · 5656 in / 1011 out tokens · 23507 ms · 2026-05-23T02:48:52.966701+00:00 · methodology

discussion (0)

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Reference graph

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