Information of spin space group $A\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:38.38.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
2a	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,c |0,0,z)
2b	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/2,0,c |0,0,z)
4c	$.\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(a,0,c |0,0,z)
4d	$\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(0,b,c |0,0,z)
4e	$\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(1/2,b,c |0,0,z)
8f	$\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}{2}}\ce{^{\infty m}{1}} $
T:(1,0,1/2)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Y:(1,0,0)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Z:(0,0,1/2)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
R:(1/2,1/2,1/2)	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $
S:(1/2,1/2,0)	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $
A:(u,0,1/2)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
B:(0,v,1/2)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
H:(1,0,w)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
D:(1/2,1/2,w)	$\ce{^{1}{2}}\ce{^{\infty}{1}} $
K:(0,v,w)	$\ce{^{1}{m}}\ce{^{\infty}{1}} $
M:(u,0,w)	$\ce{^{1}{m}}\ce{^{\infty}{1}} $
P:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
Q:(u,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

2a

k-vector	A1↑G(1)
Γ:(0,0,0)	$Γ_{1}^{S}(1) $
T:(1/2,0,-1)	$T_{1}^{S}(1) $
Y:(0,0,-1)	$Y_{1}^{S}(1) $
Z:(1/2,0,0)	$Z_{1}^{S}(1) $
R:(1/2,1/2,-1/2)	$R_{1}^{S}(1) $
S:(0,1/2,-1/2)	$S_{1}^{S}(1) $
A:(1/2,0,w)	$A_{1}^{S}(1) $
B:(1/2,v,0)	$B_{1}^{S}(1) $
Δ:(0,v,0)	$Δ_{1}^{S}(1) $
H:(u,0,-1)	$H_{1}^{S}(1) $
Λ:(u,0,0)	$Λ_{1}^{S}(1) $
Σ:(0,0,w)	$Σ_{1}^{S}(1) $
D:(u,1/2,-1/2)	$D_{1}^{S}(1) $
K:(u,v,0)	$K_{1}^{S}(1) $
M:(u,0,w)	$M_{1}^{S}(1) $
P:(0,v,w)	$P_{1}^{S}(1) $
Q:(1/2,v,w)	$Q_{1}^{S}(1) $
GP:(u,v,w)	$GP_{1}^{S}(1) $

2b

k-vector	A1↑G(1)
Γ:(0,0,0)	$Γ_{1}^{S}(1) $
T:(1/2,0,-1)	$T_{3}^{S}(1) $
Y:(0,0,-1)	$Y_{1}^{S}(1) $
Z:(1/2,0,0)	$Z_{3}^{S}(1) $
R:(1/2,1/2,-1/2)	$R_{2}^{S}(1) $
S:(0,1/2,-1/2)	$S_{1}^{S}(1) $
A:(1/2,0,w)	$A_{3}^{S}(1) $
B:(1/2,v,0)	$B_{2}^{S}(1) $
Δ:(0,v,0)	$Δ_{1}^{S}(1) $
H:(u,0,-1)	$H_{1}^{S}(1) $
Λ:(u,0,0)	$Λ_{1}^{S}(1) $
Σ:(0,0,w)	$Σ_{1}^{S}(1) $
D:(u,1/2,-1/2)	$D_{1}^{S}(1) $
K:(u,v,0)	$K_{1}^{S}(1) $
M:(u,0,w)	$M_{1}^{S}(1) $
P:(0,v,w)	$P_{1}^{S}(1) $
Q:(1/2,v,w)	$Q_{2}^{S}(1) $
GP:(u,v,w)	$GP_{1}^{S}(1) $

4c

k-vector	A'↑G(2)
Γ:(0,0,0)	$Γ_{1}^{S}(1)⊕Γ_{3}^{S}(1) $
T:(1/2,0,-1)	$T_{1}^{S}(1)⊕T_{3}^{S}(1) $
Y:(0,0,-1)	$Y_{1}^{S}(1)⊕Y_{3}^{S}(1) $
Z:(1/2,0,0)	$Z_{1}^{S}(1)⊕Z_{3}^{S}(1) $
R:(1/2,1/2,-1/2)	$R_{1}^{S}(1)⊕R_{2}^{S}(1) $
S:(0,1/2,-1/2)	$S_{1}^{S}(1)⊕S_{2}^{S}(1) $
A:(1/2,0,w)	$A_{1}^{S}(1)⊕A_{3}^{S}(1) $
B:(1/2,v,0)	$B_{1}^{S}(1)⊕B_{2}^{S}(1) $
Δ:(0,v,0)	$Δ_{1}^{S}(1)⊕Δ_{2}^{S}(1) $
H:(u,0,-1)	$2H_{1}^{S}(1) $
Λ:(u,0,0)	$2Λ_{1}^{S}(1) $
Σ:(0,0,w)	$Σ_{1}^{S}(1)⊕Σ_{3}^{S}(1) $
D:(u,1/2,-1/2)	$2D_{1}^{S}(1) $
K:(u,v,0)	$2K_{1}^{S}(1) $
M:(u,0,w)	$2M_{1}^{S}(1) $
P:(0,v,w)	$P_{1}^{S}(1)⊕P_{2}^{S}(1) $
Q:(1/2,v,w)	$Q_{1}^{S}(1)⊕Q_{2}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

4d

k-vector	A'↑G(2)
Γ:(0,0,0)	$Γ_{1}^{S}(1)⊕Γ_{4}^{S}(1) $
T:(1/2,0,-1)	$T_{1}^{S}(1)⊕T_{4}^{S}(1) $
Y:(0,0,-1)	$Y_{1}^{S}(1)⊕Y_{4}^{S}(1) $
Z:(1/2,0,0)	$Z_{1}^{S}(1)⊕Z_{4}^{S}(1) $
R:(1/2,1/2,-1/2)	$2R_{1}^{S}(1) $
S:(0,1/2,-1/2)	$2S_{1}^{S}(1) $
A:(1/2,0,w)	$A_{1}^{S}(1)⊕A_{4}^{S}(1) $
B:(1/2,v,0)	$2B_{1}^{S}(1) $
Δ:(0,v,0)	$2Δ_{1}^{S}(1) $
H:(u,0,-1)	$H_{1}^{S}(1)⊕H_{2}^{S}(1) $
Λ:(u,0,0)	$Λ_{1}^{S}(1)⊕Λ_{2}^{S}(1) $
Σ:(0,0,w)	$Σ_{1}^{S}(1)⊕Σ_{4}^{S}(1) $
D:(u,1/2,-1/2)	$2D_{1}^{S}(1) $
K:(u,v,0)	$2K_{1}^{S}(1) $
M:(u,0,w)	$M_{1}^{S}(1)⊕M_{2}^{S}(1) $
P:(0,v,w)	$2P_{1}^{S}(1) $
Q:(1/2,v,w)	$2Q_{1}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

4e

k-vector	A'↑G(2)
Γ:(0,0,0)	$Γ_{1}^{S}(1)⊕Γ_{4}^{S}(1) $
T:(1/2,0,-1)	$T_{2}^{S}(1)⊕T_{3}^{S}(1) $
Y:(0,0,-1)	$Y_{1}^{S}(1)⊕Y_{4}^{S}(1) $
Z:(1/2,0,0)	$Z_{2}^{S}(1)⊕Z_{3}^{S}(1) $
R:(1/2,1/2,-1/2)	$2R_{2}^{S}(1) $
S:(0,1/2,-1/2)	$2S_{1}^{S}(1) $
A:(1/2,0,w)	$A_{2}^{S}(1)⊕A_{3}^{S}(1) $
B:(1/2,v,0)	$2B_{2}^{S}(1) $
Δ:(0,v,0)	$2Δ_{1}^{S}(1) $
H:(u,0,-1)	$H_{1}^{S}(1)⊕H_{2}^{S}(1) $
Λ:(u,0,0)	$Λ_{1}^{S}(1)⊕Λ_{2}^{S}(1) $
Σ:(0,0,w)	$Σ_{1}^{S}(1)⊕Σ_{4}^{S}(1) $
D:(u,1/2,-1/2)	$2D_{1}^{S}(1) $
K:(u,v,0)	$2K_{1}^{S}(1) $
M:(u,0,w)	$M_{1}^{S}(1)⊕M_{2}^{S}(1) $
P:(0,v,w)	$2P_{1}^{S}(1) $
Q:(1/2,v,w)	$2Q_{2}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

8f

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S}(1)⊕Γ_{2}^{S}(1)⊕Γ_{3}^{S}(1)⊕Γ_{4}^{S}(1) $
T:(1/2,0,-1)	$T_{1}^{S}(1)⊕T_{2}^{S}(1)⊕T_{3}^{S}(1)⊕T_{4}^{S}(1) $
Y:(0,0,-1)	$Y_{1}^{S}(1)⊕Y_{2}^{S}(1)⊕Y_{3}^{S}(1)⊕Y_{4}^{S}(1) $
Z:(1/2,0,0)	$Z_{1}^{S}(1)⊕Z_{2}^{S}(1)⊕Z_{3}^{S}(1)⊕Z_{4}^{S}(1) $
R:(1/2,1/2,-1/2)	$2R_{1}^{S}(1)⊕2R_{2}^{S}(1) $
S:(0,1/2,-1/2)	$2S_{1}^{S}(1)⊕2S_{2}^{S}(1) $
A:(1/2,0,w)	$A_{1}^{S}(1)⊕A_{2}^{S}(1)⊕A_{3}^{S}(1)⊕A_{4}^{S}(1) $
B:(1/2,v,0)	$2B_{1}^{S}(1)⊕2B_{2}^{S}(1) $
Δ:(0,v,0)	$2Δ_{1}^{S}(1)⊕2Δ_{2}^{S}(1) $
H:(u,0,-1)	$2H_{1}^{S}(1)⊕2H_{2}^{S}(1) $
Λ:(u,0,0)	$2Λ_{1}^{S}(1)⊕2Λ_{2}^{S}(1) $
Σ:(0,0,w)	$Σ_{1}^{S}(1)⊕Σ_{2}^{S}(1)⊕Σ_{3}^{S}(1)⊕Σ_{4}^{S}(1) $
D:(u,1/2,-1/2)	$4D_{1}^{S}(1) $
K:(u,v,0)	$4K_{1}^{S}(1) $
M:(u,0,w)	$2M_{1}^{S}(1)⊕2M_{2}^{S}(1) $
P:(0,v,w)	$2P_{1}^{S}(1)⊕2P_{2}^{S}(1) $
Q:(1/2,v,w)	$2Q_{1}^{S}(1)⊕2Q_{2}^{S}(1) $
GP:(u,v,w)	$4GP_{1}^{S}(1) $