invariant_orthogonal_projector

invariant_orthogonal_projector(rep_x)

Computes the orthogonal projection to the invariant subspace.

The input representation \(\rho_{\mathcal{X}}: \mathbb{G} \mapsto \mathbb{G}\mathbb{L}(\mathcal{X})\) is transformed to the spectral basis given by:

\[\rho_{\mathcal{X}} = \mathbf{Q}\left( \bigoplus_{k\in[1,n_{\text{iso}}]} \bigoplus_{i\in[1,n_k]} \hat{\rho}_k \right)\mathbf{Q}^T\]

where \(\hat{\rho}_k\) are irreducible representations of \(\mathbb{G}\), \(n_k\) is the multiplicity of type \(k\), and \(\mathbf{Q}: \mathcal{X}\to\mathcal{X}\) is the orthogonal change of basis from the irrep-spectral basis to the original basis.

Define the diagonal selector \(\mathbf{S}\in\mathbb{R}^{D\times D}\) in irrep-spectral coordinates by

\[\begin{split}S_{jj} = \begin{cases} 1, & \text{if coordinate } j \text{ belongs to a trivial irrep copy}, \\ 0, & \text{otherwise}. \end{cases}\end{split}\]

Then the orthogonal projector onto the invariant subspace \(\mathcal{X}^{\text{inv}}=\{\mathbf{x}\in\mathcal{X}: \rho_{\mathcal{X}}(g)\mathbf{x}=\mathbf{x}, \forall g\in\mathbb{G}\}\) is

\[\mathbf{P}_{\mathrm{inv}} = \mathbf{Q}\,\mathbf{S}\,\mathbf{Q}^T.\]

This projector enforces the invariance constraint:

\[\rho_{\mathcal{X}}(g)\,\mathbf{P}_{\mathrm{inv}} = \mathbf{P}_{\mathrm{inv}}\,\rho_{\mathcal{X}}(g) = \mathbf{P}_{\mathrm{inv}} \quad \forall g\in\mathbb{G}.\]

Parameters:

rep_x (Representation) – The representation for which the orthogonal projection to the invariant subspace is computed.

Returns:

The orthogonal projection matrix to the invariant subspace, \(\mathbf{Q} \mathbf{S} \mathbf{Q}^T\).

Return type:

Tensor