is_complex_irreducible

is_complex_irreducible(group, rep)

Check complex irreducibility of \(\rho:\mathbb{G}\to\mathrm{GL}(\mathcal{X})\).

By Schur’s lemma, \(\rho\) is complex-irreducible iff every Hermitian matrix commuting with all \(\rho(g)\) is scalar. The routine searches for a non-scalar commuting Hermitian witness \(\mathbf{H}\).

Parameters:

• group (Group) – Symmetry group of the representation.

• rep (dict | collections.abc.Callable) – A Union-style input. It must be either a dict or a collections.abc.Callable. It must map each GroupElement to a matrix ndarray.

Returns:

• (True, I) if irreducible (identity witness),

• (False, H) if reducible with a non-scalar commuting Hermitian witness.

Return type:

tuple[bool, ndarray]