Hall column
The Hall entry lists the symmetry-allowed components of a time-reversal-odd axial vector \(\mathbf{h}=(h_x,h_y,h_z)\). A zero vector means that no Hall-vector component is permitted for that MPG.
Net T-tensor notation
The table uses net third-rank tensor components \(T_{\alpha\beta\gamma}\). The first two indices are the symmetric spatial pair, \(T_{\alpha\beta\gamma}=T_{\beta\alpha\gamma}\), and the third index is the spin index. This notation deliberately avoids a factorised on-site form: such a factorisation applies to an individual on-site contribution, whereas the table describes the total tensor obtained after summing contributions over all sublattices.
All 18 component labels
T-vector and T-tensor columns
The T-vector column shows the supplied symmetry-reduced tensor components directly. Multiple terms in the same row are separated by simple + or − signs. The T-tensor column shows the supplied net tensor-component relations.
The T-vector expressions are clickable in the guide, results table, and group-detail view. Selecting one filters the table to groups whose T-vector column contains that component.
The T-vector column
The nine T-vector component labels satisfy
\(\gamma=\alpha\qquad\text{or}\qquad\gamma=\beta\).
Within each table cell, T-vector components are grouped by their free index. Terms belonging to the same row are written as a simple expression such as \(T_{xxx}+T_{xyy}+T_{xzz}\). Because the first two tensor indices are symmetric, the stored canonical component \(T_{xyx}\) belongs to the free-\(y\) row.
The T-tensor column
The T-tensor column reports the net third-rank tensor components and any coefficient/equality relations supplied by the data. Components are grouped by the third (spin) index, so \(T_{\alpha\beta x}\), \(T_{\alpha\beta y}\), and \(T_{\alpha\beta z}\) appear on their corresponding \(S_x\), \(S_y\), and \(S_z\) rows in the clicked detail view.
To reduce crowding, the compact table view omits T-tensor entries that are already shown in the T-vector column for the same row. The omitted count is still shown. Enable Show all terms in table to display the full 3 × 6 tensor layout in the source index order. Each allowed component stays in its fixed index position, forbidden components remain visible as faint empty pill placeholders, equality/sign relations are listed separately below the component grid, and dashed pills mark entries that also appear under T-vector.
Primes and overbars
A prime denotes combination with time reversal. An overbar denotes rotoinversion notation, for example \(\bar{4}\). Groups containing the explicit element \(1^{\prime}\) are highlighted in the detail view.
Change of basis
The change-of-basis tool accepts either an axis–angle rotation or a directly entered basis matrix. Axis–angle input uses \(M_{\mathrm{new}}=R^{\mathsf T}M_{\mathrm{old}}R\); direct matrix input uses \(M_{\mathrm{new}}=C^{-1}M_{\mathrm{old}}C\).
The direct matrix fields let you work without first converting a basis change into a rotation axis and angle. In the comparison workspace, the shared editor applies either the axis–angle input or the direct matrix input to the active comparison view.
The updated source data gives net tensor components with explicit coefficients and component relations in the original basis, and those are shown directly in the table. The change-of-basis display treats them as rank-3 tensor components and propagates support term by term; it does not transform the supplied relations as a coupled invariant subspace or re-solve the magnetic-point-group null space.
Compare bases opens the dedicated comparison workspace in a separate tab. Select the two-, three-, or four-panel icon, choose a basis for each view, and mark the basis you want as active. Use Apply active basis to explorer table to send it back to the main table.
In the Compare Bases tab, applying a basis to the explorer table confirms that the basis was applied, then starts a five-second window-closing countdown. Select Cancel if you want to keep the comparison open.
What this table does not show
The original-basis table now preserves the explicit coefficients, equalities, sign changes, and ratios present in the supplied data. It does not, however, replace the full magnetic-point-group null-space calculation or explicitly parameterise the complete invariant space in terms of independent material coefficients.