Semi-Invariants of a Matrix and Covector

Jonathan Elmer

The SL_d-invariants of a matrix and a covector form a polynomial ring generated by the characteristic polynomial coefficients and one determinant Δ.

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2026-07-15 03:42 UTC pith:PPV6JHL7

load-bearing objection Clean freeness theorem for SL_d on a matrix plus a covector, with one explicit extra generator; solid classical statement, body not inspectable here. the 2 major comments →

arxiv 2607.12738 v1 pith:PPV6JHL7 submitted 2026-07-14 math.AC

Semi-Invariants of a Matrix and Covector

Jonathan Elmer This is my paper

classification math.AC MSC 13A5016G2015A72

keywords invariant theoryspecial linear groupmatrix conjugationcharacteristic polynomialsemi-invariantsquiver representationspolynomial ringcovector

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that when the special linear group SL_d acts on the space of d-by-d matrices together with a row vector by simultaneous conjugation and right multiplication, the ring of polynomial invariants is free. It is generated by the d coefficients of the characteristic polynomial of the matrix plus one additional invariant Δ, the determinant of the d-by-d matrix whose rows are the covector successively multiplied by powers of the matrix. A sympathetic reader cares because this gives a complete, explicit description of all polynomial functions constant on the orbits, without having to compute a full generating set from scratch. The argument is classical and works over any infinite field; the author also reinterprets the same statement in the language of quiver representations. The result therefore supplies a clean structural fact about a basic mixed representation of SL_d.

Core claim

For the natural SL_d-action on the affine space of pairs (A, φ) consisting of a d-by-d matrix and a covector, the invariant ring K[X]^{SL_d} is a polynomial ring freely generated by the coefficients of the characteristic polynomial of A together with the single additional invariant Δ(A,φ)=det(φ,φA,…,φA^{d-1})^t.

What carries the argument

The determinant invariant Δ(A,φ), formed by stacking the successive images of the covector under powers of the matrix and taking the determinant of the resulting square matrix; it supplies the single extra generator needed once the characteristic polynomial coefficients are known.

Load-bearing premise

The base field must be infinite; the classical freeness and generation arguments used in the paper rely on this hypothesis and may fail over finite fields.

What would settle it

Over an infinite field, produce a homogeneous SL_d-invariant of the pair (A,φ) that cannot be written as a polynomial in the characteristic coefficients and Δ, or exhibit an algebraic relation among those generators.

Watch this falsifier — get emailed when new claim-graph text bears on it.

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

Clean freeness theorem for SL_d on a matrix plus a covector, with one explicit extra generator; solid classical statement, body not inspectable here.

read the letter

The one thing to know: Elmer proves that the SL_d-invariants of a d×d matrix together with a covector form a polynomial ring, generated by the usual characteristic-polynomial coefficients of the matrix plus one extra explicit invariant Δ = det(φ, φA, …, φA^{d-1})^t. That freeness statement with the concrete generator is the actual new result; pure conjugation invariants are classical, and adjoining the covector is the increment.

What it does well is state a precise, usable theorem. The generators are written down, the action is standard, and the abstract promises a classical proof plus a quiver-representation reading. That packaging is useful for people who already think in terms of quiver moduli or semi-invariants; it does not invent a new framework, but it does give a clean freeness result that sits naturally next to the classical char-poly generators.

Soft spots are mostly about what we cannot see. Only the abstract is available, so the algebraic-independence argument and the classical generation steps are not checkable. The infinite-field hypothesis is load-bearing in the usual way for these arguments; over finite fields the same generators may fail. That is a standard limitation, not a hidden flaw, but it should be kept in view. Nothing in the abstract looks circular or invented.

This is for classical invariant theorists and for people working on quiver representations who want an explicit free ring in a concrete SL_d setting. It is not a reorganizing paper, but it is the sort of clean, citable lemma that gets used. I would send it to a serious referee rather than desk-reject; the claim is sharp enough and the generators explicit enough to deserve a full check of the proof. If the body holds up, it is a solid within-subfield contribution.

Referee Report

2 major / 0 minor

Summary. The manuscript asserts that, for an infinite field K, the SL_d-invariant ring of X = M_d ⊕ (K^d)* under the action g·(A,φ) = (gAg^{-1}, φg^{-1}) is a polynomial algebra, freely generated by the d coefficients of the characteristic polynomial of A together with one further invariant Δ(A,φ) := det(φ, φA, …, φA^{d-1})^t. The argument is described as classical, with an additional interpretation in the language of quiver representations.

Significance. An explicit freeness theorem for this natural SL_d-action would be a clean, usable result in classical invariant theory and in the representation theory of quivers. The generators are concrete (characteristic-polynomial coefficients plus a single determinant Δ), and a fully classical proof would make the statement accessible as a reference theorem. The claimed algebraic independence and generation, if established rigorously, constitute a solid contribution.

major comments (2)

Only the abstract is available for this review; the classical generation argument and the algebraic-independence argument for the proposed generators are therefore not inspectable. The central freeness claim cannot be verified or refuted on the present evidence, so the load-bearing steps remain unchecked.
Abstract: the hypothesis that K is infinite is stated explicitly and is load-bearing for standard classical arguments (orbit density, non-vanishing of resultants/discriminants, etc.). The manuscript must clarify whether generation or freeness fails over finite fields, or supply the precise characteristic restrictions under which the proof operates.

Circularity Check

0 steps flagged

No significant circularity; abstract-only classical freeness claim is self-contained as stated.

full rationale

Only the abstract is available. It states a classical freeness theorem for the SL_d-invariant ring of matrices plus a covector: K[X]^{SL_d} is polynomial, generated by the coefficients of the characteristic polynomial of A together with the single extra invariant Δ defined by an explicit determinant formula. No fitted parameters, no self-definitional reduction of generators to the claimed freeness, no uniqueness theorem imported from the authors, and no ansatz smuggled via self-citation appear in the given text. The sole explicit hypothesis is that K is infinite, which is standard for classical generation/independence arguments and is not circular. The quiver-representation interpretation is presented as optional commentary, not as a load-bearing premise. With no body available, no further derivation steps can be inspected; on the material provided the derivation is self-contained and non-circular. Score 0 is therefore the honest finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Pure commutative-algebra theorem over an infinite field. No numerical parameters are fitted. The only non-standard input is the infiniteness hypothesis on K; everything else is standard linear algebra and classical invariant theory. No new geometric or physical entities are postulated.

axioms (2)

domain assumption K is an infinite field
Stated in the abstract; classical generation arguments for invariant rings typically require infinite base fields to guarantee enough specializations.
standard math Standard facts on characteristic polynomials and determinants under conjugation
Used to identify the obvious generators; classical and independent of the new freeness claim.

pith-pipeline@v1.1.0-grok45 · 6056 in / 1764 out tokens · 20631 ms · 2026-07-15T03:42:46.418411+00:00 · methodology

0 comments

read the original abstract

We prove the following theorem: let $\mathcal{M}_d$ denote the set of $d \times d$ matrices over an infinite field $K$, and let ${(K^d)^*}$ be the set of row vectors. Define an action of $\mathrm{SL}_d(K)$ on $X:= \mathcal{M}_d \oplus (K^d)^*$ by \[ g \cdot (A,\phi) = (gAg^{-1}, \phi g^{-1}).\] Then $K[X]^{\mathrm{SL}_d}$ is a polynomial ring, generated by the coefficients of the characteristic polynomial of $A$ and one further invariant, namely $\Delta(A,\phi):= \det(\phi,\phi A,\phi A^2,\ldots, \phi A^{d-1})^t.$ Our proof is entirely classical in nature, but we give an interpretation of the result and its proof in terms of quiver representation theory.

Semi-Invariants of a Matrix and Covector

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)