Information of spin space group $P\ce{^{1}{\text{-}4}}\ce{^{\infty m}{1}}$ ( nSSG:81.81.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-a, -b, c | x, y, z	{ 1 ‖ 2001 | 0 }
b, -a, -c | x, y, z	{ 1 ‖ -4+001 | 0 }
-b, a, -c | x, y, z	{ 1 ‖ -4-001 | 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
b, -a, -c | -x, y, z	{ m ‖ -4+001 | 0 }
-b, a, -c | -x, y, z	{ m ‖ -4-001 | 0 }

Wyckoff Positions

WP	Site symmetry	Representative
1a	$\ce{^{1}{-4}}..\ce{^{\infty m}{1}} $	(0,0,0 |0,0,z)
1b	$\ce{^{1}{-4}}..\ce{^{\infty m}{1}} $	(0,0,1/2 |0,0,z)
1c	$\ce{^{1}{-4}}..\ce{^{\infty m}{1}} $	(1/2,1/2,0 |0,0,z)
1d	$\ce{^{1}{-4}}..\ce{^{\infty m}{1}} $	(1/2,1/2,1/2 |0,0,z)
2e	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(0,0,c |0,0,z)
2f	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(1/2,1/2,c |0,0,z)
2g	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(0,1/2,c |0,0,z)
4h	$\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
A:(1/2,1/2,1/2)	$\ce{^{1}{-4}}\ce{^{\infty m}{1}} $
Γ:(0,0,0)	$\ce{^{1}{-4}}\ce{^{\infty m}{1}} $
M:(1/2,1/2,0)	$\ce{^{1}{-4}}\ce{^{\infty m}{1}} $
Z:(0,0,1/2)	$\ce{^{1}{-4}}\ce{^{\infty m}{1}} $
R:(0,1/2,1/2)	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $
X:(0,1/2,0)	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Λ:(0,0,w)	$\ce{^{m}{-4}}\ce{^{\infty}{1}} $
V:(1/2,1/2,w)	$\ce{^{m}{-4}}\ce{^{\infty}{1}} $
D:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
E:(u,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
S:(u,u,1/2)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
Σ:(u,u,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
T:(u,1/2,1/2)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
U:(0,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
W:(0,1/2,w)	$\ce{^{1}{2}}\ce{^{\infty}{1}} $
Y:(u,1/2,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
B:(0,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $
C:(u,u,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $
F:(u,1/2,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

1a

k-vector	A↑G(1)
A:(1/2,1/2,1/2)	$A_{1}^{S}(1) $
Γ:(0,0,0)	$Γ_{1}^{S}(1) $
M:(1/2,1/2,0)	$M_{1}^{S}(1) $
Z:(0,0,1/2)	$Z_{1}^{S}(1) $
R:(0,1/2,1/2)	$R_{1}^{S}(1) $
X:(0,1/2,0)	$X_{1}^{S}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1) $
V:(1/2,1/2,w)	$V_{1}^{S}(1) $
D:(u,v,0)	$D_{1}^{S}(1) $
Δ:(0,v,0)	$Δ_{1}^{S}(1) $
E:(u,v,1/2)	$E_{1}^{S}(1) $
S:(u,u,1/2)	$S_{1}^{S}(1) $
Σ:(u,u,0)	$Σ_{1}^{S}(1) $
T:(u,1/2,1/2)	$T_{1}^{S}(1) $
U:(0,v,1/2)	$U_{1}^{S}(1) $
W:(0,1/2,w)	$W_{1}^{S}(1) $
Y:(u,1/2,0)	$Y_{1}^{S}(1) $
B:(0,v,w)	$B_{1}^{S}(1) $
C:(u,u,w)	$C_{1}^{S}(1) $
F:(u,1/2,w)	$F_{1}^{S}(1) $
GP:(u,v,w)	$GP_{1}^{S}(1) $

1b

k-vector	A↑G(1)
A:(1/2,1/2,1/2)	$A_{2}^{S}(1) $
Γ:(0,0,0)	$Γ_{1}^{S}(1) $
M:(1/2,1/2,0)	$M_{1}^{S}(1) $
Z:(0,0,1/2)	$Z_{2}^{S}(1) $
R:(0,1/2,1/2)	$R_{1}^{S}(1) $
X:(0,1/2,0)	$X_{1}^{S}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1) $
V:(1/2,1/2,w)	$V_{1}^{S}(1) $
D:(u,v,0)	$D_{1}^{S}(1) $
Δ:(0,v,0)	$Δ_{1}^{S}(1) $
E:(u,v,1/2)	$E_{1}^{S}(1) $
S:(u,u,1/2)	$S_{1}^{S}(1) $
Σ:(u,u,0)	$Σ_{1}^{S}(1) $
T:(u,1/2,1/2)	$T_{1}^{S}(1) $
U:(0,v,1/2)	$U_{1}^{S}(1) $
W:(0,1/2,w)	$W_{1}^{S}(1) $
Y:(u,1/2,0)	$Y_{1}^{S}(1) $
B:(0,v,w)	$B_{1}^{S}(1) $
C:(u,u,w)	$C_{1}^{S}(1) $
F:(u,1/2,w)	$F_{1}^{S}(1) $
GP:(u,v,w)	$GP_{1}^{S}(1) $

1c

k-vector	A↑G(1)
A:(1/2,1/2,1/2)	$A_{2}^{S}(1) $
Γ:(0,0,0)	$Γ_{1}^{S}(1) $
M:(1/2,1/2,0)	$M_{2}^{S}(1) $
Z:(0,0,1/2)	$Z_{1}^{S}(1) $
R:(0,1/2,1/2)	$R_{2}^{S}(1) $
X:(0,1/2,0)	$X_{2}^{S}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1) $
V:(1/2,1/2,w)	$V_{1}^{S}(1) $
D:(u,v,0)	$D_{1}^{S}(1) $
Δ:(0,v,0)	$Δ_{1}^{S}(1) $
E:(u,v,1/2)	$E_{1}^{S}(1) $
S:(u,u,1/2)	$S_{1}^{S}(1) $
Σ:(u,u,0)	$Σ_{1}^{S}(1) $
T:(u,1/2,1/2)	$T_{1}^{S}(1) $
U:(0,v,1/2)	$U_{1}^{S}(1) $
W:(0,1/2,w)	$W_{2}^{S}(1) $
Y:(u,1/2,0)	$Y_{1}^{S}(1) $
B:(0,v,w)	$B_{1}^{S}(1) $
C:(u,u,w)	$C_{1}^{S}(1) $
F:(u,1/2,w)	$F_{1}^{S}(1) $
GP:(u,v,w)	$GP_{1}^{S}(1) $

1d

k-vector	A↑G(1)
A:(1/2,1/2,1/2)	$A_{1}^{S}(1) $
Γ:(0,0,0)	$Γ_{1}^{S}(1) $
M:(1/2,1/2,0)	$M_{2}^{S}(1) $
Z:(0,0,1/2)	$Z_{2}^{S}(1) $
R:(0,1/2,1/2)	$R_{2}^{S}(1) $
X:(0,1/2,0)	$X_{2}^{S}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1) $
V:(1/2,1/2,w)	$V_{1}^{S}(1) $
D:(u,v,0)	$D_{1}^{S}(1) $
Δ:(0,v,0)	$Δ_{1}^{S}(1) $
E:(u,v,1/2)	$E_{1}^{S}(1) $
S:(u,u,1/2)	$S_{1}^{S}(1) $
Σ:(u,u,0)	$Σ_{1}^{S}(1) $
T:(u,1/2,1/2)	$T_{1}^{S}(1) $
U:(0,v,1/2)	$U_{1}^{S}(1) $
W:(0,1/2,w)	$W_{2}^{S}(1) $
Y:(u,1/2,0)	$Y_{1}^{S}(1) $
B:(0,v,w)	$B_{1}^{S}(1) $
C:(u,u,w)	$C_{1}^{S}(1) $
F:(u,1/2,w)	$F_{1}^{S}(1) $
GP:(u,v,w)	$GP_{1}^{S}(1) $

2e

k-vector	A↑G(2)
A:(1/2,1/2,1/2)	$A_{1}^{S}(1)⊕A_{2}^{S}(1) $
Γ:(0,0,0)	$Γ_{1}^{S}(1)⊕Γ_{2}^{S}(1) $
M:(1/2,1/2,0)	$M_{1}^{S}(1)⊕M_{2}^{S}(1) $
Z:(0,0,1/2)	$Z_{1}^{S}(1)⊕Z_{2}^{S}(1) $
R:(0,1/2,1/2)	$2R_{1}^{S}(1) $
X:(0,1/2,0)	$2X_{1}^{S}(1) $
Λ:(0,0,w)	$2Λ_{1}^{S}(1) $
V:(1/2,1/2,w)	$2V_{1}^{S}(1) $
D:(u,v,0)	$2D_{1}^{S}(1) $
Δ:(0,v,0)	$2Δ_{1}^{S}(1) $
E:(u,v,1/2)	$2E_{1}^{S}(1) $
S:(u,u,1/2)	$2S_{1}^{S}(1) $
Σ:(u,u,0)	$2Σ_{1}^{S}(1) $
T:(u,1/2,1/2)	$2T_{1}^{S}(1) $
U:(0,v,1/2)	$2U_{1}^{S}(1) $
W:(0,1/2,w)	$2W_{1}^{S}(1) $
Y:(u,1/2,0)	$2Y_{1}^{S}(1) $
B:(0,v,w)	$2B_{1}^{S}(1) $
C:(u,u,w)	$2C_{1}^{S}(1) $
F:(u,1/2,w)	$2F_{1}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

2f

k-vector	A↑G(2)
A:(1/2,1/2,1/2)	$A_{1}^{S}(1)⊕A_{2}^{S}(1) $
Γ:(0,0,0)	$Γ_{1}^{S}(1)⊕Γ_{2}^{S}(1) $
M:(1/2,1/2,0)	$M_{1}^{S}(1)⊕M_{2}^{S}(1) $
Z:(0,0,1/2)	$Z_{1}^{S}(1)⊕Z_{2}^{S}(1) $
R:(0,1/2,1/2)	$2R_{2}^{S}(1) $
X:(0,1/2,0)	$2X_{2}^{S}(1) $
Λ:(0,0,w)	$2Λ_{1}^{S}(1) $
V:(1/2,1/2,w)	$2V_{1}^{S}(1) $
D:(u,v,0)	$2D_{1}^{S}(1) $
Δ:(0,v,0)	$2Δ_{1}^{S}(1) $
E:(u,v,1/2)	$2E_{1}^{S}(1) $
S:(u,u,1/2)	$2S_{1}^{S}(1) $
Σ:(u,u,0)	$2Σ_{1}^{S}(1) $
T:(u,1/2,1/2)	$2T_{1}^{S}(1) $
U:(0,v,1/2)	$2U_{1}^{S}(1) $
W:(0,1/2,w)	$2W_{2}^{S}(1) $
Y:(u,1/2,0)	$2Y_{1}^{S}(1) $
B:(0,v,w)	$2B_{1}^{S}(1) $
C:(u,u,w)	$2C_{1}^{S}(1) $
F:(u,1/2,w)	$2F_{1}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

2g

k-vector	A↑G(2)
A:(1/2,1/2,1/2)	$A_{3}^{S}A_{4}^{S}(2) $
Γ:(0,0,0)	$Γ_{1}^{S}(1)⊕Γ_{2}^{S}(1) $
M:(1/2,1/2,0)	$M_{3}^{S}M_{4}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(1)⊕Z_{2}^{S}(1) $
R:(0,1/2,1/2)	$R_{1}^{S}(1)⊕R_{2}^{S}(1) $
X:(0,1/2,0)	$X_{1}^{S}(1)⊕X_{2}^{S}(1) $
Λ:(0,0,w)	$2Λ_{1}^{S}(1) $
V:(1/2,1/2,w)	$V_{2}^{S}V_{2}^{S}(2) $
D:(u,v,0)	$2D_{1}^{S}(1) $
Δ:(0,v,0)	$2Δ_{1}^{S}(1) $
E:(u,v,1/2)	$2E_{1}^{S}(1) $
S:(u,u,1/2)	$2S_{1}^{S}(1) $
Σ:(u,u,0)	$2Σ_{1}^{S}(1) $
T:(u,1/2,1/2)	$2T_{1}^{S}(1) $
U:(0,v,1/2)	$2U_{1}^{S}(1) $
W:(0,1/2,w)	$W_{1}^{S}(1)⊕W_{2}^{S}(1) $
Y:(u,1/2,0)	$2Y_{1}^{S}(1) $
B:(0,v,w)	$2B_{1}^{S}(1) $
C:(u,u,w)	$2C_{1}^{S}(1) $
F:(u,1/2,w)	$2F_{1}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

4h

k-vector	A↑G(4)
A:(1/2,1/2,1/2)	$A_{1}^{S}(1)⊕A_{2}^{S}(1)⊕A_{3}^{S}A_{4}^{S}(2) $
Γ:(0,0,0)	$Γ_{1}^{S}(1)⊕Γ_{2}^{S}(1)⊕Γ_{3}^{S}Γ_{4}^{S}(2) $
M:(1/2,1/2,0)	$M_{1}^{S}(1)⊕M_{2}^{S}(1)⊕M_{3}^{S}M_{4}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(1)⊕Z_{2}^{S}(1)⊕Z_{3}^{S}Z_{4}^{S}(2) $
R:(0,1/2,1/2)	$2R_{1}^{S}(1)⊕2R_{2}^{S}(1) $
X:(0,1/2,0)	$2X_{1}^{S}(1)⊕2X_{2}^{S}(1) $
Λ:(0,0,w)	$2Λ_{1}^{S}(1)⊕Λ_{2}^{S}Λ_{2}^{S}(2) $
V:(1/2,1/2,w)	$2V_{1}^{S}(1)⊕V_{2}^{S}V_{2}^{S}(2) $
D:(u,v,0)	$4D_{1}^{S}(1) $
Δ:(0,v,0)	$4Δ_{1}^{S}(1) $
E:(u,v,1/2)	$4E_{1}^{S}(1) $
S:(u,u,1/2)	$4S_{1}^{S}(1) $
Σ:(u,u,0)	$4Σ_{1}^{S}(1) $
T:(u,1/2,1/2)	$4T_{1}^{S}(1) $
U:(0,v,1/2)	$4U_{1}^{S}(1) $
W:(0,1/2,w)	$2W_{1}^{S}(1)⊕2W_{2}^{S}(1) $
Y:(u,1/2,0)	$4Y_{1}^{S}(1) $
B:(0,v,w)	$4B_{1}^{S}(1) $
C:(u,u,w)	$4C_{1}^{S}(1) $
F:(u,1/2,w)	$4F_{1}^{S}(1) $
GP:(u,v,w)	$4GP_{1}^{S}(1) $