Minimum dimension of determinant-minus-one finite-order matrices

Which dimensions N admit an integer matrix of a prescribed finite order m and determinant −1? Define ψ(m) as the sum of φ(pᵏ) over the exact prime-power factors of m, omitting only an exact factor 2, not higher powers of 2; the target criterion is m even and N≥ψ(m)+1.

Why it matters and what counts as progress

Why it matters: Prescribing determinant −1 adds a structural constraint missing from the classical classification of finite-order rational and integral matrices.

What counts as progress: Prove the least-dimension formula and the matching rational-matrix criterion or determine a sharper classification with further determinant constraints.

How the problem developed

1928 · Classical result [1]
Vaidyanathaswamy Recorded the classical finite-order matrix criterion underlying the later determinant-minus-one extension.
10 Oct 2026 · Criterion [1]
GPT-6 Pro, with Claude credited for an independent derivation and verification Derived the determinant-minus-one minimum-dimension criterion for even orders and supplied its formal proof account.

AI contribution

AI-derived determinant restriction for finite-order matrices

For m>0 and N≥1, the formalized refinement says GL(N,ℤ) contains an element of order m with determinant −1 exactly when m is even and ψ(m)+1≤N, where ψ sums φ(p^k) over exact prime-power factors except 2. The same criterion holds over ℚ, and the least integral dimension is ψ(m)+1.

AI involvementAI-led

ImportanceModest

Technical verificationFormal (Lean 4/Mathlib kernel with transitive axiom audit)

Human understandingFew