60.22 WyckoffLattice

WyckoffLattice( G )

If a point x in a Wyckoff position W_1 has a stabilizer which is a subgroup of the stabilizer of some point y in a Wyckoff position W_2, then the closure of W_1 will contain W_2. These incidence relations are best represented in a graph. WyckoffLattice( G ) determines and displays this graph using XGAP (note that XGAP runs only under Unix plus the X Window System). Each Wyckoff position is represented by a vertex. If W_1 contains W_2, its vertex is placed below that of W_2 (i.e., Wyckoff positions with bigger stabilizers are placed higher up), and the two are connected, either directly (if there is no other Wyckoff position in between) or indirectly. With the left mouse button and with the XGAP CleanUp menu it is possible to change the layout of the graph (see the XGAP manual). When clicking with the right mouse button on a vertex, a pop up menu appears, which allows to obtain the following information about the representative affine subspace of the Wyckoff position:

StabDim:

Dimension of the affine subspace of stable points.

StabSize:

Size of the stabilizer subgroup.

ClassSize:

Number of Wyckoff positions having a stabilizer whose point group is in the same subgroup conjugacy class.

IsAbelian, IsCyclic, IsNilpotent, IsPerfect, IsSimple, IsSolvable:

Information about the stabilizer subgroup.

Isomorphism:

Isomorphism type of the stabilizer subgroup. Works only for small sizes.

ConjClassInfo:

Prints (in the GAP window) information about each of the conjugacy classes of the stabilizer, namely the order, the trace and the determinant of its elements, and the size of the conjugacy class. Note that trace refers here only to the trace of the point group part, without the trailing 1 of the affine matrix.

Translation:

The representative point of the affine subspace.

Basis:

The basis of the linear space parallel to the affine subspace.

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