Symmetry-Preserving Spectral Projector Iterations in Exact and Approximate Representation Commutants

This work develops a symmetry-preserving framework for computing spectral projectors inside exact and approximate commutants of finite-dimensional group representations. Two classical projector iterations are considered: McWeeny polynomial purification and the Roberts–Newton iteration for the matrix sign function. Their equivariance-preserving properties, convergence mechanisms, and use in recursive representation decomposition are analyzed. For approximately equivariant data, we introduce the commutator constraint operator and a conditioned commutant gap that converts measurable commutator residuals into bounds on the distance from the exact representation commutant. The degenerate case in which the commutant equals the full endomorphism algebra is treated explicitly. For noisy representation generators, fully computable a posteriori certificates are derived from observed singular values and declared perturbation bounds. These estimates are combined with singular-subspace perturbation theory and spectral-projector bounds to control both the recovered approximate commutant and the resulting invariant subspaces. An observable stopping floor is obtained for finite McWeeny and Newton matrix-sign iterations. The paper distinguishes classical ingredients—including McWeeny purification, the Roberts–Newton matrix-sign iteration, Reynolds averaging, and standard singular- and spectral-subspace perturbation results—from the proposed integration of these tools into a certified workflow for exact and approximate representation commutants. Reproducible numerical experiments illustrate the predicted perturbation regime and the behavior of the computable certificates.