The strong Lefschetz property for complex reflection coinvariants

The ordinary coinvariant algebra over C of every finite complex reflection group has the strong Lefschetz property: one linear element has maximal-rank multiplication by every power in every degree. We give a computer-assisted proof for G₃₄, the exceptional type excluded by Wachi's earlier results; the other types follow from the work of Harima–Watanabe, Harima–Migliore–Nagel–Watanabe, McDaniel, and Wachi. The question is recorded in Wachi's 2013 conjecture on complex reflection coinvariants. The proof uses the Conway–Sloane orbit invariants in the coordinates developed by Oura–Sekiguchi. Exact finite-field polynomial certificates, specialization, and a Hilbert-function comparison establish the strong Lefschetz property for a relative coinvariant algebra. The free-extension theorem then transfers it to the ordinary coinvariant algebra, using Wachi's result for G₃₃. Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.