LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Trace-coordinate algebra for the SL2 character quotient

Statement

Assume full AC and let $G$ be a finitely presented group. Every regular function on $Hom (G, S L_{2} (C))$ invariant under simultaneous conjugation is a polynomial in finitely many word-trace functions $t_{w} (ρ) = tr (ρ (w))$ . Equivalently, the coordinate algebra of the affine character quotient is generated by trace functions of words in $G$ .

Statement references

Procesi, The invariant theory of $n \times n$ matrices, Theorem 1.3

Tillmann, §§4.2–4.8

Facts & Assumptions

Given: Full AC, a finite presentation of $G$ , its affine representation variety, and the simultaneous conjugation action.

[F1]

For any finite presentation, the determinant and adjugate relator equations give the actual closed invariant representation subset in the matrix tuple space, including its reduced coordinate algebra. This general finitely presented group statement is proved directly in SL2 matrix invariants from epsilon contractions and traces, not restricted to knot groups.

[F2]

Every word trace is a regular conjugation-invariant function (Trace functions are regular and conjugation invariant).

[F3]

The invariant coordinate algebra is finitely generated and defines the affine categorical quotient (Finite generation of invariants and the affine categorical quotient).

[F4]

SL2 matrix invariants from epsilon contractions and traces proves trace generation for the full simultaneous matrix action by explicit epsilon tensor contractions and polarization, and proves the passage to every closed invariant reduced subset using the actual reductive quotient theorem.

Proof

1.1F1givenconstruct

Use the finite presentation of $G$ to realize its representations as the determinant-one matrix tuples satisfying the relator equations from [F1]. This is a closed invariant reduced subset of the finite matrix space; no assumption that $G$ is a knot group is needed.

2.1F2F3F4step 1.1algebra∎

Apply [F4] to that actual subset. Every invariant regular function is a polynomial in finitely many restricted word traces. On determinant-one matrices an adjugate in that calculation is the inverse matrix, so the words are words in $G$ , and their regular invariant functions agree with [F2]. Thus the invariant coordinate algebra is generated by the stated trace functions. The full-AC quotient branch is exactly the branch of [F3] and [F4], rather than an unproved assertion of exactness for the relator ideal.

Depends on

Used by

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. Procesi, The invariant theory of n x n matrices, Advances in Mathematics 19 (1976), Theorem 1.3 (standard reference, not scraped)
Stephan Tillmann, Varieties associated to 3-manifolds, §§4.2–4.8 (standard reference, not scraped)