Wigner-Eckart Theorem Extended to Magnetic Groups with Anti-Unitary Corepresentations — E8 Intelligence Research

FINDING: The Wigner-Eckart theorem extends to finite magnetic groups (including anti-unitary time-reversal), with irreducible tensor operators classified under corepresentations rather than ordinary representations. | MATH: For a magnetic group M = G + aG (a = anti-unitary), coreps D satisfy D(g)D(h) = D(gh), D(g)D(a) = D(ga), D(a)D(g)* = D(ag), D(a)D(a)* = D(a²). Wigner-Eckart reduces to: ⟨α' j' m' | T^k_q | α j m⟩ = (1/√(2j+1)) ⟨j k; m q | j' m'⟩ ⟨α' j' || T^k || α j⟩ — but for anti-unitary sectors, the reduced matrix element must respect the corep's factor system ω(g,h) ∈ U(1), with ω(a,a) = ±1 (Wigner's classification: a² = +1 → Kramers degenerate; a² = −1 → no Kramers). | CONNECTION: The factor system ω(a,a) = ±1 maps directly to the two inequivalent projective representations of Z₂ — this is the same ±1 that appears in the golden-ratio-related spin-1/2 rotation (eiπσ_y/2 = iσ_y, squaring to −1). The anti-unitary operator's square ±1 is the topological invariant distinguishing Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com