NL Layer Groups

After general pages (tables of groups, generators and Brillouin zones) for each cluster of groups (set of single and double ordinary and gray groups; to be completed by black-and-white ones) chosen in the menu, there is a page offering list of available datasets for every group of the cluster. These data are given by a gif image of a table (and a picture in some cases) with necessary explanations of entries. Bottom bar enables to switch between properties or to magnify each table.

Generators are given by the symbols introduced in the first column, and explained in the second one; their crystallographic symbols are in the third column, and, when appropriate, conditions for some group parameters are in the last column. More detailed specification is obtained by the subscripts: a and b are lattice basis vectors from the xy-plane, vector q is from the xy-plane and it defines mirror plane perpendicular to it or rotational axis along it.

Symbol	Symetry operation	Int. symbol
C_n	Rotation for 2π/n around z-axis	n₀₀₁	n=1,2,3,4,6
U, U_q	Rotation for π around horizontal q-axis	2_q	q={q₁,q₂,0}
σ_v, σ_{v_q}	Vertical mirror plane	m_q	q={q₁,q₂,0}
σ_h	Horizontal mirror plane	m₀₀₁
J,S₂	Spatial inversion	1
S_2n	C_2nσ_h	S_2n	n=1, 2, 3
t	Translation	(1\|t)	t={t₁,t₂,0}
t_h	Horizontal glide plane	(m₀₀₁\|1/2t)	t∈{a, b, a+b}
t_{v_q}	Vertical glide plane	(m_q\|1/2t)	t∈{a, b, a+b}, (t⊥q)
t_{U_q}	Screw axis	(2_q\|1/2t), (2₁)	t∈{a, b, a+b}, (t∥q)
θ	Time reversal	T

As for the strata (generic, special lines and points) of the Brillouin zone, there are several sets of labels in the literature. We did not find conceptual justification for either of them. This is due to possibility to derive the irreducible representations from the space supergroups and succeed their strata labels; however, the strata symbols for the space groups are not systematic, and this confusion is further strengthen by non-uniqueness of the choice of space supergroup.

Therefore we give the following set of single letter symbols. Generic stratum is G, Greek letters are special lines, and Latin capitals for special point (except traditional Γ point). The line along k₁ is Δ and parallel to it Ω, the line along k₂ is Φ and parallel to it Λ, while Σ and Ξ are the bisectors of the angles ∠(k₁,k₂) and ∠(k₁,-k₂). Bellow is parallel list of the here applied and Bilbao server notation.

Oblique lattice
Label	Coordinates	Bilbao
Γ	{0,0}	Γ
X	{π,0}	B
Y	{0,π}	Y
S	{π,π}	A

Orthorhombic lattice
Label	Coordinates	Bilbao
Γ	{0,0}	Γ
X	{π,0}	X
Y	{0,π}	Y
S	{π,π}	S
Δ	{k₁,0}	Σ
Ω	{k₁,π}	C
Φ	{0,k₂}	Δ
Λ	{π,k₂}	D

Rhombic lattice
Label	Coordinates	Bilbao
Γ	{0,0}	Γ
Y	{0,π}	S
S	{π,π}	Y
Σ	{k,k}	Σ
Ξ	{-k,k}	Δ

Square lattice
Label	Coordinates	Bilbao
Γ	{0,0}	Γ
X	{π,0}	X
S	{π,π}	M
Δ	{k₁,0}	Δ
Σ	{k,k}	Σ
Λ	{π,k₂}	Y

Hexagonal lattice
Label	Coordinates	Bilbao
Γ	{0,0}	Γ
K	{2π/3,2π/3}	K
L	{4π/3,-2π/3}	KA
Δ	{k₁,0}	Σ
Φ	{0,k₂}	SN
Σ	{k,k}	Λ
Ξ	{k,-k/2}	LE
Λ	{k₁,-2k₂+2π}	T, TA

O - Orbit: set of points such that group elements map one into another.
|O| - Orbit order: number of orbit points per unit cell.
F_x - Fixing group (Stabilizer, Little group) of x: subgroup fixing point x.
REP - (co-)representation
IR - irreducible (co-)representation
ID - irreducible domain
Factorization - each layer group is product of a generalized translational group and an axial point group, G=T̂P. Single group is weak direct product.
Factorization generators: Single layer group is weak direct product of the cyclic subgroups and monomial form of elements:
G=C(g₁)···C(g_[G]), g=(g₁)^α₁···(g_[G])^α_(G),
with bi-unique exponents α_i. Generators are ordered: g₁,g₂ - generalized translations (infinnite order), g₃=C_n or S_2n (orders n or 2n), g₄,... "parities" (order 2). In double groups, ẽ′ of order two (antipodal of the identity) comes after geometrical transformations; also, the exponents are bi-unique provided their range as in single group.
In gray groups the last generator is time reversal θ.
Grading generators: For each layer group G=G^(G) there is a sequence of layer groups G⁽ⁱ⁾ such that the form of the first is G⁽¹⁾=TC_n, while G⁽ⁱ⁾ is halving subgroup in G⁽ⁱ⁺¹⁾: G⁽ⁱ⁺¹⁾=G⁽ⁱ⁾+ℓ_i+3G⁽ⁱ⁾ with (ℓ_i+3)² from G⁽ⁱ⁾. Each element is expanded as monomial in graded generators:
g=(ℓ₁)^β₁···(ℓ_(G))^β_(G), with exponents β₁,β₂=0,±1,…; β₃=0,…,n-1; β₄,..., β_(G)=0,1.
Maximal Wickoff positions: orbits with maximal stabilizers (which are not subgroups of any other stabilizer).
Euler angles (for M from O(3)) {φ,θ,ψ,det(M)}: det(M)=±1 for M from/out of SO(3), three angles in x-convention

Notation, terminology, abbreviations