Bilbao Crystallographic Server CoRepresentations

# Notes on the notation of the co-representations in the Bilbao Crystallographic Server

The co-representations of a magnetic group are obtained from the irreducible representations of its unitary subgroup. For the notation of the co-representations of the little groups, the label of the co-representation is obtained from the label (or labels) of the corresponding irreducible representation(s). In those cases (in some type III groups) in which the label of the k-vector in the magnetic group is different from the label in the unitary subgroup, the notation includes as a prefix the label of the magnetic group.
In the tables of co-representations and in the following examples, the BNS setting is assumed.

## Examples of the notation used for the co-representations of the little groups

### Example 1: magnetic group Cmc211' (N. 36.173)

It is a magnetic group of type II (gray group), and its unitary subgroup is the space group Cmc21 (N. 36).
• At point Γ:(0,0,0) in the reciprocal space of the magnetic group, the irreps are obtained from the irreps at the Γ point of the reciprocal space of the space group. All the irreps Γ1, Γ2, Γ3, Γ4 and Γ5 are of type (a), and they induce the co-representations with the same label.

• At point T:(1,0,1/2) of the magnetic group, the co-representations are obtained from the representations at point T:(1,0,1/2) of the space group.

The pair of irreps T1 and T3 are of type (c), and they induce the correpresentation T1T3 in Cmc211'.

The pair of irreps T2 and T4 are of type (c), and they induce the correpresentation T2T4 in Cmc211'.

The double-valued irrep T5 is of type (b), and it induces the correpresentation T5T5 in Cmc211'.

• At point H:(1,0,w) of the magnetic group, the co-representations are obtained from the representations at point H:(1,0,w) of the space group.

All the irreps (H1, H2, H3, H4 and H5) are of type (x), i.e., there are no antiunitary operations in the magnetic group that transform H:(1,0,w) into -(1,0,w) (mod translations of the reciprocal lattice). The labels of the co-representations are (H1, H2, H3, H4 and H5).

### Example 2: magnetic group P4'21'm (N. 113.269)

It is a magnetic group of type III whose unitary subgroup is the space group Cmm2 (N. 33). The unitary operations are not in the standard setting of the space group. The transformation matrix to the standard setting is (1/2a-1/b,1/2a+1/2b,c;0,1/2,0)
• At point F:(u,1/2,w) of the magnetic group, the co-representations are obtained from the representations at point GP:(1/2+u,1/2-u,w) of the space group in its standard setting. The two irreps GP1 and GP2 are of type (b). The induced co-representations are denoted as (F)GP1GP1 and (F)GP2GP2

Using this notation, on the one hand, the label gives information about the origin of the co-representation (the labels of the irreps of the unitary subgroup) and, on the other hand, the prefix indicates the specific point in the reciprocal space of the magnetic group.

• At point B:(0,v,w) of the magnetic group, the co-representations are obtained from the representations at point GP:(v,v,w) of the space group. The two irreps GP1 and GP2 are of type (a). The induced co-representations are denoted as (B)GP1 and (B)GP2

• At point GP:(u,v,w) of the magnetic group, the co-representations are obtained from the representations at point GP:(u+v,-u+v,w) of the space group. The two irreps GP1 and GP2 are of type (x). The induced co-representations are denoted as GP1 and GP2.

In this last case, as the labels in the magnetic group and in the unitary subgroup are the same, no prefix is added.
Note that, in this example, if the prefix is omited, the resulting labels of the co-representations could be misleading.

## Examples of the notation used for the full co-representations

### Example 3: magnetic group I4' (N. 79.27)

It is a magnetic group of type III whose unitary subgroup is the space group C2 (N. 5). The unitary operations are not in the standard setting of the space group. The transformation matrix to the standard setting is (a-c,a,-b;0,0,0)

• In the white group I4, the point X in the reciprocal space has two branches in the star, X:(1/2,1/2,0),(1/2,-1/2,0). These points correspond to the points M:(0,1,1/2) and A:(0,0,1/2), respectively, in the standard setting of its space group C2. Therefore, the co-representations of the little group of X:(1/2,1/2,0) in the magnetic group are obtained from the irreps at M:(0,1,1/2) in the unitary subgroup and the co-representations at X:(1/2,-1/2,0) from the irreps at A:(0,0,1/2).

The co-representations of the little group are denoted as (X)M1, (X)M2, (X)M3 and (X)A4 in the first case and as (X)A1, (X)A2, (X)A3 and (X)A4 in the second one.

The full representations at X are denoted as,

*(X)M1A1, *(X)M2A2, *(X)M3A4 and *(X)M4A3.

Note that the pairs of conjugated irreps that join to form the full co-representations are (M1,A1), (M2,A2), (M3,A4) and (M4,A3).