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.
Tensor-product notation
A component such as \(z^2S_x\) is displayed as \(Q_{zz}S_x\). Likewise, \(xzS_y\) means \(Q_{xz}S_y\). The app treats \(Q_{\alpha\beta}\) as a symmetric rank-2 label.
All 18 component labels
T-vector and T-tensor columns
The table divides the 18 products \(Q_{\alpha\beta}S_\gamma\) into two sets of nine.
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 T-vector set consists of terms for which the spin index is also one of the two \(Q\) indices:
\(\gamma=\alpha\qquad\text{or}\qquad\gamma=\beta\).
The T-tensor column
The T-tensor column contains the complementary nine terms for which the spin index is not one of the two \(Q\) indices:
\(\gamma\neq\alpha,\qquad\gamma\neq\beta\).
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 those overlaps; dashed T-tensor 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. The preset list includes a \(45^\circ\) rotation about \(z\) and the hexagonal-to-Cartesian matrix used in the Python analysis.
Because this table stores component support rather than the invariant-space coefficient relations, the tool transforms each listed tensor component separately. It does not re-solve the magnetic-point-group null space.
The regular Change basis tool opens in the explorer tab. Use Compare bases inside the change-of-basis sidebar to open a separate comparison tab, select the two-, three-, or four-panel icon, and choose a basis for each view. Mark the basis you want as active, then 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 table lists component support, not null-space coefficients. It therefore does not establish whether two displayed components are independent, equal, opposite in sign, or constrained by another linear relation.