Symmetry actions and sectors

qslib stores a site permutation as source_for_destination. Its action is

[ (g\cdot b)j=b{p_g(j)}, ]

so it has the same gather direction as a row-major array read. Permutation validates bijectivity, composes and inverts actions, and can act on dense binary states or map Pauli supports. A separately named adapter is required when converting to a destination-for-source convention.

translation and translation_group derive coordinate actions from the declared geometry and boundaries. Rectangle and square point-group builders return validated finite groups. Open-boundary translations that leave the finite domain are rejected; periodic translations wrap explicitly.

SpinInversion applies (b_i\mapsto1-b_i). It is an available action, not an automatic claim that a particular Hamiltonian is invariant. Use is_interaction_symmetry to verify that every resolved weighted interaction, including its channel, name, bond, and coefficient, is preserved. validate_model_symmetry checks the complete resolved Pauli Hamiltonian, including onsite fields, and reports the first missing mapped term; validate_spin_inversion performs the corresponding commutator check.

DiagonalGauge implements (\phi(b)=\pi\sum_i a_i b_i) and applies the resulting complex phase to amplitudes. Integer coefficients are classified as sign-only. sublattice_gauge supplies the checkerboard gauge for square geometries and rejects unsupported triangular embeddings.

For a finite group and a one-dimensional character, FiniteGroup::project_amplitudes implements

[ (P_\chi\psi)(b)=|G|^{-1}\sum_g\chi(g)^*\psi(g^{-1}\cdot b). ]

Orbit methods return deterministic packed-state representatives. Projection and orbit utilities operate on explicit dense amplitudes and are intended as small-system reference primitives before sector-specific exact solvers.