Information of spin space group $P\ce{^{\text{-}1}{m}}\ce{^{\text{-}1}{m}}\ce{^{\text{-}1}{a}}\ce{^{\infty m}{1}}$ ( nSSG:17.51.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
2a	$.\ce{^{1}{2}}/\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
2b	$.\ce{^{1}{2}}/\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,1/2,0 | 0,0,0)
2c	$.\ce{^{1}{2}}/\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,0)
2d	$.\ce{^{1}{2}}/\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,1/2,1/2 | 0,0,0)
2e	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,0,c | 0,0,0)
2f	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/2,c | 0,0,0)
4g	$.\ce{^{1}{2}}.\ce{^{\infty m}{1}} $	(0,b,0 | 0,0,z)
4h	$.\ce{^{1}{2}}.\ce{^{\infty m}{1}} $	(0,b,1/2 | 0,0,z)
4i	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,0,c | 0,0,0)
4j	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,1/2,c | 0,0,0)
4k	$\ce{^{-1}{m}}..\ce{^{\infty m}{1}} $	(1/4,b,c | 0,0,0)
8l	$\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{^{2}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
B:(0,v,1/2)	$ \ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
C:(u,1/2,0)	$ \ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
D:(1/2,v,0)	$ \ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$ \ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
E:(u,1/2,1/2)	$ \ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
G:(1/2,0,w)	$ \ce{^{2}{m}}\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
H:(0,1/2,w)	$ \ce{^{2}{m}}\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{2}{m}}\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
P:(1/2,v,1/2)	$ \ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Q:(1/2,1/2,w)	$ \ce{^{2}{m}}\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$ \ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{2}{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{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
N:(u,1/2,w)	$ \ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
V:(u,v,0)	$ \ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
W:(u,v,1/2)	$ \ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$ \ce{^{-1}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4g

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

4h

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

8l

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