Information of spin space group $P\ce{^{\text{-}1}{m}}\ce{^{1}{m}}\ce{^{1}{n}}\ce{^{\infty m}{1}}$ ( nSSG:31.59.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
a+1/2, -b, -c | x, y, z	{ 1 ‖ 2100 | 1/2 0 0 }
a, -b+1/2, c | x, y, z	{ 1 ‖ m010 | 0 1/2 0 }
a+1/2, b+1/2, -c | x, y, z	{ 1 ‖ m001 | 1/2 1/2 0 }
-a, b+1/2, -c | -x, -y, -z	{ -1 ‖ 2010 | 0 1/2 0 }
-a+1/2, -b+1/2, c | -x, -y, -z	{ -1 ‖ 2001 | 1/2 1/2 0 }
-a, -b, -c | -x, -y, -z	{ -1 ‖ -1 | 0 }
-a+1/2, b, c | -x, -y, -z	{ -1 ‖ m100 | 1/2 0 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
a+1/2, -b, -c | -x, y, z	{ m ‖ 2100 | 1/2 0 0 }
a, -b+1/2, c | -x, y, z	{ m ‖ m010 | 0 1/2 0 }
a+1/2, b+1/2, -c | -x, y, z	{ m ‖ m001 | 1/2 1/2 0 }
-a, b+1/2, -c | x, -y, -z	{ 2 ‖ 2010 | 0 1/2 0 }
-a+1/2, -b+1/2, c | x, -y, -z	{ 2 ‖ 2001 | 1/2 1/2 0 }
-a, -b, -c | x, -y, -z	{ 2 ‖ -1 | 0 }
-a+1/2, b, c | x, -y, -z	{ 2 ‖ m100 | 1/2 0 0 }

Wyckoff Positions

WP	Site symmetry	Representative
2a	$\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,c | 0,0,0)
2b	$\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,3/4,c | 0,0,0)
4c	$\ce{^{-1}{-1}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
4d	$\ce{^{-1}{-1}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,0)
4e	$\ce{^{-1}{m}}..\ce{^{\infty m}{1}} $	(1/4,b,c | 0,0,0)
4f	$.\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(a,1/4,c | 0,0,z)
8g	$\ce{^{1}{1}}\ce{^{\infty m}{1}} $	(a,b,c | 0,0,z)

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group
Γ:(0,0,0)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
R:(1/2,1/2,1/2)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
S:(1/2,1/2,0)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
T:(0,1/2,1/2)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
U:(1/2,0,1/2)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
X:(1/2,0,0)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
Y:(0,1/2,0)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
Z:(0,0,1/2)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
A:(u,0,1/2)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
B:(0,v,1/2)	$ \ce{^{2}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
C:(u,1/2,0)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
D:(1/2,v,0)	$ \ce{^{2}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$ \ce{^{2}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
E:(u,1/2,1/2)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
G:(1/2,0,w)	$ \ce{^{2}{m}}\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
H:(0,1/2,w)	$ \ce{^{2}{m}}\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{2}{m}}\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
P:(1/2,v,1/2)	$ \ce{^{2}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Q:(1/2,1/2,w)	$ \ce{^{2}{m}}\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
K:(0,v,w)	$ \ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
L:(1/2,v,w)	$ \ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
M:(u,0,w)	$ \ce{^{-1}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
N:(u,1/2,w)	$ \ce{^{-1}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
V:(u,v,0)	$ \ce{^{-1}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
W:(u,v,1/2)	$ \ce{^{-1}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$ \ce{^{-1}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4f

k-vector	k-vector-G↑	A'↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{3}^{S}(2) $
R:(1/2,-1/2,-1/2)	R:(1/2,1/2,1/2)	$R_{1}^{S}(4) $
S:(1/2,-1/2,0)	U:(1/2,0,1/2)	$(S)U_{1}^{S}(4) $
T:(0,-1/2,-1/2)	S:(1/2,1/2,0)	$(T)S_{1}^{S}(4) $
U:(1/2,0,-1/2)	T:(0,1/2,1/2)	$(U)T_{1}^{S}T_{3}^{S}(4) $
X:(1/2,0,0)	Z:(0,0,1/2)	$(X)Z_{1}^{S}Z_{3}^{S}(4) $
Y:(0,-1/2,0)	X:(1/2,0,0)	$(Y)X_{1}^{S}(4) $
Z:(0,0,-1/2)	Y:(0,1/2,0)	$(Z)Y_{1}^{S}(2)⊕(Z)Y_{3}^{S}(2) $
A:(u,0,-1/2)	H:(0,1/2,u)	$(A)H_{1}^{S}(2)⊕(A)H_{3}^{S}(2) $
B:(0,v,-1/2)	C:(-v,1/2,0)	$(B)C_{1}^{S}(2)⊕(B)C_{2}^{S}(2) $
C:(u,-1/2,0)	G:(1/2,0,u)	$(C)G_{1}^{S}(4) $
D:(1/2,v,0)	A:(-v,0,1/2)	$(D)A_{1}^{S}A_{2}^{S}(4) $
Δ:(0,v,0)	Σ:(-v,0,0)	$(Δ)Σ_{1}^{S}(2)⊕(Δ)Σ_{2}^{S}(2) $
E:(u,-1/2,-1/2)	Q:(1/2,1/2,u)	$(E)Q_{1}^{S}(4) $
G:(1/2,0,w)	B:(0,-w,1/2)	$(G)B_{1}^{S}B_{1}^{S}(4) $
H:(0,-1/2,w)	D:(1/2,-w,0)	$(H)D_{1}^{S}D_{2}^{S}(4) $
Λ:(0,0,w)	Δ:(0,-w,0)	$2(Λ)Δ_{1}^{S}(2) $
P:(1/2,v,-1/2)	E:(-v,1/2,1/2)	$(P)E_{1}^{S}E_{2}^{S}(4) $
Q:(1/2,-1/2,w)	P:(1/2,-w,1/2)	$(Q)P_{1}^{S}P_{2}^{S}(4) $
Σ:(u,0,0)	Λ:(0,0,u)	$(Σ)Λ_{1}^{S}(2)⊕(Σ)Λ_{3}^{S}(2) $
K:(0,v,w)	V:(-v,-w,0)	$2(K)V_{1}^{S}(2) $
L:(1/2,v,w)	W:(-v,-w,1/2)	$(L)W_{1}^{S}W_{1}^{S}(4) $
M:(u,0,w)	K:(0,-w,u)	$2(M)K_{1}^{S}(2) $
N:(u,-1/2,w)	L:(1/2,-w,u)	$(N)L_{1}^{S}(2)⊕(N)L_{2}^{S}(2) $
V:(u,v,0)	M:(-v,0,u)	$(V)M_{1}^{S}(2)⊕(V)M_{2}^{S}(2) $
W:(u,v,-1/2)	N:(-v,1/2,u)	$(W)N_{1}^{S}(2)⊕(W)N_{2}^{S}(2) $
GP:(u,v,w)	GP:(-v,-w,u)	$2GP_{1}^{S}(2) $

8g

k-vector	k-vector-G↑	A↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2)⊕Γ_{3}^{S}(2)⊕Γ_{4}^{S}(2) $
R:(1/2,-1/2,-1/2)	R:(1/2,1/2,1/2)	$2R_{1}^{S}(4) $
S:(1/2,-1/2,0)	U:(1/2,0,1/2)	$2(S)U_{1}^{S}(4) $
T:(0,-1/2,-1/2)	S:(1/2,1/2,0)	$2(T)S_{1}^{S}(4) $
U:(1/2,0,-1/2)	T:(0,1/2,1/2)	$(U)T_{1}^{S}T_{3}^{S}(4)⊕(U)T_{2}^{S}T_{4}^{S}(4) $
X:(1/2,0,0)	Z:(0,0,1/2)	$(X)Z_{1}^{S}Z_{3}^{S}(4)⊕(X)Z_{2}^{S}Z_{4}^{S}(4) $
Y:(0,-1/2,0)	X:(1/2,0,0)	$2(Y)X_{1}^{S}(4) $
Z:(0,0,-1/2)	Y:(0,1/2,0)	$(Z)Y_{1}^{S}(2)⊕(Z)Y_{2}^{S}(2)⊕(Z)Y_{3}^{S}(2)⊕(Z)Y_{4}^{S}(2) $
A:(u,0,-1/2)	H:(0,1/2,u)	$(A)H_{1}^{S}(2)⊕(A)H_{2}^{S}(2)⊕(A)H_{3}^{S}(2)⊕(A)H_{4}^{S}(2) $
B:(0,v,-1/2)	C:(-v,1/2,0)	$2(B)C_{1}^{S}(2)⊕2(B)C_{2}^{S}(2) $
C:(u,-1/2,0)	G:(1/2,0,u)	$2(C)G_{1}^{S}(4) $
D:(1/2,v,0)	A:(-v,0,1/2)	$2(D)A_{1}^{S}A_{2}^{S}(4) $
Δ:(0,v,0)	Σ:(-v,0,0)	$2(Δ)Σ_{1}^{S}(2)⊕2(Δ)Σ_{2}^{S}(2) $
E:(u,-1/2,-1/2)	Q:(1/2,1/2,u)	$2(E)Q_{1}^{S}(4) $
G:(1/2,0,w)	B:(0,-w,1/2)	$(G)B_{1}^{S}B_{1}^{S}(4)⊕(G)B_{2}^{S}B_{2}^{S}(4) $
H:(0,-1/2,w)	D:(1/2,-w,0)	$2(H)D_{1}^{S}D_{2}^{S}(4) $
Λ:(0,0,w)	Δ:(0,-w,0)	$2(Λ)Δ_{1}^{S}(2)⊕2(Λ)Δ_{2}^{S}(2) $
P:(1/2,v,-1/2)	E:(-v,1/2,1/2)	$2(P)E_{1}^{S}E_{2}^{S}(4) $
Q:(1/2,-1/2,w)	P:(1/2,-w,1/2)	$2(Q)P_{1}^{S}P_{2}^{S}(4) $
Σ:(u,0,0)	Λ:(0,0,u)	$(Σ)Λ_{1}^{S}(2)⊕(Σ)Λ_{2}^{S}(2)⊕(Σ)Λ_{3}^{S}(2)⊕(Σ)Λ_{4}^{S}(2) $
K:(0,v,w)	V:(-v,-w,0)	$4(K)V_{1}^{S}(2) $
L:(1/2,v,w)	W:(-v,-w,1/2)	$2(L)W_{1}^{S}W_{1}^{S}(4) $
M:(u,0,w)	K:(0,-w,u)	$2(M)K_{1}^{S}(2)⊕2(M)K_{2}^{S}(2) $
N:(u,-1/2,w)	L:(1/2,-w,u)	$2(N)L_{1}^{S}(2)⊕2(N)L_{2}^{S}(2) $
V:(u,v,0)	M:(-v,0,u)	$2(V)M_{1}^{S}(2)⊕2(V)M_{2}^{S}(2) $
W:(u,v,-1/2)	N:(-v,1/2,u)	$2(W)N_{1}^{S}(2)⊕2(W)N_{2}^{S}(2) $
GP:(u,v,w)	GP:(-v,-w,u)	$4GP_{1}^{S}(2) $