# Superspace point group

### From Online Dictionary of Crystallography

Groupe ponctuel de superespace (*Fr*). Punktgruppe des Superraums (*Ge*). Gruppo puntuale di superspazio (*It*). 超空間の点群 (*Ja*). Grupo puntual del superespacio (*Sp*).

## Definition

An (*m+d*)-dimensional superspace group is a space group with a point group *K* that leaves
an *m*-dimensional (real) subspace invariant. Therefore, *K* is R-reducible and its elements are
pairs () of orthogonal transformations. Both *R*_{E} and *R*_{I} may themselves be R-reducible in turn. They form the *m*-dimensional point group *K*_{E}, and the *d*-dimensional point group *K*_{I}, respectively.

## Comments

On a lattice basis the point group elements are represented by integral matrices Γ(*R*).
The action of the point group on the reciprocal lattice is given by the integral matrix Γ^{ * }(*R*), which is the inverse transpose of Γ(*R*).

The diffraction spots of an aperiodic crystal belong to a vector module *M*^{ * } that is the
projection of the *n*-dimensional reciprocal lattice Σ^{ * } on the physical space. The projections
of the basis vectors of Σ^{ * } are the basis vectors of the vector module *M*^{ * }. Therefore,
the action of the *n*-dimensional point group of the superspace group on the basis of *M*^{ * } is

For an incommensurate modulated structure, the submodule of the main reflections is invariant. As a consequence, the elements of the point group in superspace in this case is Z-reducible. There is a basis such that the point group elements are represented by the integral matrices

Both and are integral representations of *K*,
as are their conjugates Γ_{E}(*K*) and Γ_{I}(*K*).

Points in direct space, with lattice coordinates transform according to

In direct space the internal space *V*_{I} is left invariant, and this subspace contains a *d*-dimensional lattice, that is left invariant.