Information of spin space group $P\ce{^{1}{\text{-}3}}\ce{^{1}{m}}\ce{^{1}{1}}\ce{^{\text{-}1}{(0 \ 0 \ 1/2)}}\ce{^{\infty m}{1}}$ ( nSSG:164.164.2.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-b, a-b, c | x, y, z	{ 1 ‖ 3+001 | 0 }
-a+b, -a, c | x, y, z	{ 1 ‖ 3-001 | 0 }
a-b, -b, -c | x, y, z	{ 1 ‖ 2100 | 0 }
b, a, -c | x, y, z	{ 1 ‖ 2110 | 0 }
-a, -a+b, -c | x, y, z	{ 1 ‖ 2010 | 0 }
-a, -b, -c | x, y, z	{ 1 ‖ -1 | 0 }
b, -a+b, -c | x, y, z	{ 1 ‖ -3+001 | 0 }
a-b, a, -c | x, y, z	{ 1 ‖ -3-001 | 0 }
-a+b, b, c | x, y, z	{ 1 ‖ m100 | 0 }
-b, -a, c | x, y, z	{ 1 ‖ m110 | 0 }
a, a-b, c | x, y, z	{ 1 ‖ m010 | 0 }
a, b, c+1/2 | -x, -y, -z	{ -1 ‖ 1 | 0 0 1/2 }
-b, a-b, c+1/2 | -x, -y, -z	{ -1 ‖ 3+001 | 0 0 1/2 }
-a+b, -a, c+1/2 | -x, -y, -z	{ -1 ‖ 3-001 | 0 0 1/2 }
a-b, -b, -c+1/2 | -x, -y, -z	{ -1 ‖ 2100 | 0 0 1/2 }
b, a, -c+1/2 | -x, -y, -z	{ -1 ‖ 2110 | 0 0 1/2 }
-a, -a+b, -c+1/2 | -x, -y, -z	{ -1 ‖ 2010 | 0 0 1/2 }
-a, -b, -c+1/2 | -x, -y, -z	{ -1 ‖ -1 | 0 0 1/2 }
b, -a+b, -c+1/2 | -x, -y, -z	{ -1 ‖ -3+001 | 0 0 1/2 }
a-b, a, -c+1/2 | -x, -y, -z	{ -1 ‖ -3-001 | 0 0 1/2 }
-a+b, b, c+1/2 | -x, -y, -z	{ -1 ‖ m100 | 0 0 1/2 }
-b, -a, c+1/2 | -x, -y, -z	{ -1 ‖ m110 | 0 0 1/2 }
a, a-b, c+1/2 | -x, -y, -z	{ -1 ‖ m010 | 0 0 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-b, a-b, c | -x, y, z	{ m ‖ 3+001 | 0 }
-a+b, -a, c | -x, y, z	{ m ‖ 3-001 | 0 }
a-b, -b, -c | -x, y, z	{ m ‖ 2100 | 0 }
b, a, -c | -x, y, z	{ m ‖ 2110 | 0 }
-a, -a+b, -c | -x, y, z	{ m ‖ 2010 | 0 }
-a, -b, -c | -x, y, z	{ m ‖ -1 | 0 }
b, -a+b, -c | -x, y, z	{ m ‖ -3+001 | 0 }
a-b, a, -c | -x, y, z	{ m ‖ -3-001 | 0 }
-a+b, b, c | -x, y, z	{ m ‖ m100 | 0 }
-b, -a, c | -x, y, z	{ m ‖ m110 | 0 }
a, a-b, c | -x, y, z	{ m ‖ m010 | 0 }
a, b, c+1/2 | x, -y, -z	{ 2 ‖ 1 | 0 0 1/2 }
-b, a-b, c+1/2 | x, -y, -z	{ 2 ‖ 3+001 | 0 0 1/2 }
-a+b, -a, c+1/2 | x, -y, -z	{ 2 ‖ 3-001 | 0 0 1/2 }
a-b, -b, -c+1/2 | x, -y, -z	{ 2 ‖ 2100 | 0 0 1/2 }
b, a, -c+1/2 | x, -y, -z	{ 2 ‖ 2110 | 0 0 1/2 }
-a, -a+b, -c+1/2 | x, -y, -z	{ 2 ‖ 2010 | 0 0 1/2 }
-a, -b, -c+1/2 | x, -y, -z	{ 2 ‖ -1 | 0 0 1/2 }
b, -a+b, -c+1/2 | x, -y, -z	{ 2 ‖ -3+001 | 0 0 1/2 }
a-b, a, -c+1/2 | x, -y, -z	{ 2 ‖ -3-001 | 0 0 1/2 }
-a+b, b, c+1/2 | x, -y, -z	{ 2 ‖ m100 | 0 0 1/2 }
-b, -a, c+1/2 | x, -y, -z	{ 2 ‖ m110 | 0 0 1/2 }
a, a-b, c+1/2 | x, -y, -z	{ 2 ‖ m010 | 0 0 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
2a	$\ce{^{1}{-3}}\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
2b	$\ce{^{-1}{-3}}\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(0,0,1/4 | 0,0,0)
4c	$\ce{^{1}{3}}\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(0,0,c | 0,0,z)
4d	$\ce{^{1}{3}}\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(1/3,2/3,c | 0,0,z)
6e	$.\ce{^{1}{2}}/\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(1/2,0,0 | 0,0,z)
6f	$.\ce{^{-1}{2}}/\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(1/2,0,1/4 | 0,0,0)
12g	$.\ce{^{1}{2}}.\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,z)
12h	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(a,0,1/4 | 0,0,0)
12i	$.\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(a,-a,c | 0,0,z)
24j	$\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:(0,0,1/2)	$\ce{^{1}{-3}}\ce{^{1}{m}}\ce{^{1}{1}}\ce{^{\infty /mm}{1}} $
Γ:(0,0,0)	$\ce{^{1}{-3}}\ce{^{1}{m}}\ce{^{1}{1}}\ce{^{\infty /mm}{1}} $
H:(1/3,1/3,1/2)	$\ce{^{m}{-3}}\ce{^{m}{m}}\ce{^{1}{1}}\ce{^{\infty 2}{1}} $
K:(1/3,1/3,0)	$\ce{^{m}{-3}}\ce{^{m}{m}}\ce{^{1}{1}}\ce{^{\infty 2}{1}} $
L:(1/2,0,1/2)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
M:(1/2,0,0)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
Δ:(0,0,w)	$\ce{^{m}{-3}}\ce{^{1}{m}}\ce{^{1}{1}}\ce{^{\infty 2}{1}} $
P:(1/3,1/3,w)	$\ce{^{m}{-3}}\ce{^{\infty 2}{1}} $
Λ:(u,u,0)	$\ce{^{1}{2/}}\ce{^{m}{m}}\ce{^{\infty 2}{1}} $
Σ:(u,0,0)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
Q:(u,u,1/2)	$\ce{^{1}{2/}}\ce{^{m}{m}}\ce{^{\infty 2}{1}} $
R:(u,0,1/2)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
U:(1/2,0,w)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
B:(u,v,0)	$\ce{^{m}{-1}}\ce{^{\infty 2}{1}} $
C:(u,u,w)	$\ce{^{m}{-1}}\ce{^{\infty 2}{1}} $
D:(u,0,w)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
E:(u,v,1/2)	$\ce{^{m}{-1}}\ce{^{\infty 2}{1}} $
GP:(u,v,w)	$\ce{^{m}{-1}}\ce{^{\infty 2}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

2a

k-vector	A1g↑G(2)
A:(0,0,1/2)	$A_{1}^{S,+}(2) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2) $
H:(1/3,1/3,1/2)	$H_{1}^{S}(2) $
K:(1/3,1/3,0)	$K_{1}^{S}(2) $
L:(1/2,0,1/2)	$L_{1}^{S,+}(2) $
M:(1/2,0,0)	$M_{1}^{S,+}(2) $
Δ:(0,0,w)	$Δ_{1}^{S}(2) $
P:(1/3,1/3,w)	$P_{1}^{S}(2) $
Λ:(u,u,0)	$Λ_{1}^{S}(2) $
Σ:(u,0,0)	$Σ_{1}^{S}(2) $
Q:(u,u,1/2)	$Q_{1}^{S}(2) $
R:(u,0,1/2)	$R_{1}^{S}(2) $
U:(1/2,0,w)	$U_{1}^{S}(2) $
B:(u,v,0)	$B_{1}^{S}(2) $
C:(u,u,w)	$C_{1}^{S}(2) $
D:(u,0,w)	$D_{1}^{S}(2) $
E:(u,v,1/2)	$E_{1}^{S}(2) $
GP:(u,v,w)	$GP_{1}^{S}(2) $

4c

k-vector	A1↑G(4)
A:(0,0,1/2)	$A_{1}^{S,+}(2)⊕A_{2}^{S,-}(2) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{2}^{S,-}(2) $
H:(1/3,1/3,1/2)	$H_{1}^{S}(2)⊕H_{2}^{S}(2) $
K:(1/3,1/3,0)	$K_{1}^{S}(2)⊕K_{2}^{S}(2) $
L:(1/2,0,1/2)	$L_{1}^{S,+}(2)⊕L_{2}^{S,-}(2) $
M:(1/2,0,0)	$M_{1}^{S,+}(2)⊕M_{2}^{S,-}(2) $
Δ:(0,0,w)	$2Δ_{1}^{S}(2) $
P:(1/3,1/3,w)	$2P_{1}^{S}(2) $
Λ:(u,u,0)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
Σ:(u,0,0)	$2Σ_{1}^{S}(2) $
Q:(u,u,1/2)	$Q_{1}^{S}(2)⊕Q_{2}^{S}(2) $
R:(u,0,1/2)	$2R_{1}^{S}(2) $
U:(1/2,0,w)	$2U_{1}^{S}(2) $
B:(u,v,0)	$2B_{1}^{S}(2) $
C:(u,u,w)	$2C_{1}^{S}(2) $
D:(u,0,w)	$2D_{1}^{S}(2) $
E:(u,v,1/2)	$2E_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

4d

k-vector	A1↑G(4)
A:(0,0,1/2)	$A_{1}^{S,+}(2)⊕A_{2}^{S,-}(2) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{2}^{S,-}(2) $
H:(1/3,1/3,1/2)	$H_{3}^{S}(4) $
K:(1/3,1/3,0)	$K_{3}^{S}(4) $
L:(1/2,0,1/2)	$L_{1}^{S,+}(2)⊕L_{2}^{S,-}(2) $
M:(1/2,0,0)	$M_{1}^{S,+}(2)⊕M_{2}^{S,-}(2) $
Δ:(0,0,w)	$2Δ_{1}^{S}(2) $
P:(1/3,1/3,w)	$P_{2}^{S}P_{3}^{S}(4) $
Λ:(u,u,0)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
Σ:(u,0,0)	$2Σ_{1}^{S}(2) $
Q:(u,u,1/2)	$Q_{1}^{S}(2)⊕Q_{2}^{S}(2) $
R:(u,0,1/2)	$2R_{1}^{S}(2) $
U:(1/2,0,w)	$2U_{1}^{S}(2) $
B:(u,v,0)	$2B_{1}^{S}(2) $
C:(u,u,w)	$2C_{1}^{S}(2) $
D:(u,0,w)	$2D_{1}^{S}(2) $
E:(u,v,1/2)	$2E_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

6e

k-vector	Ag↑G(6)
A:(0,0,1/2)	$A_{1}^{S,+}(2)⊕A_{3}^{S,+}(4) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{3}^{S,+}(4) $
H:(1/3,1/3,1/2)	$H_{1}^{S}(2)⊕H_{3}^{S}(4) $
K:(1/3,1/3,0)	$K_{1}^{S}(2)⊕K_{3}^{S}(4) $
L:(1/2,0,1/2)	$L_{1}^{S,+}(2)⊕L_{1}^{S,-}(2)⊕L_{2}^{S,-}(2) $
M:(1/2,0,0)	$M_{1}^{S,+}(2)⊕M_{1}^{S,-}(2)⊕M_{2}^{S,-}(2) $
Δ:(0,0,w)	$Δ_{1}^{S}(2)⊕Δ_{3}^{S}(4) $
P:(1/3,1/3,w)	$P_{1}^{S}(2)⊕P_{2}^{S}P_{3}^{S}(4) $
Λ:(u,u,0)	$2Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
Σ:(u,0,0)	$2Σ_{1}^{S}(2)⊕Σ_{2}^{S}(2) $
Q:(u,u,1/2)	$2Q_{1}^{S}(2)⊕Q_{2}^{S}(2) $
R:(u,0,1/2)	$2R_{1}^{S}(2)⊕R_{2}^{S}(2) $
U:(1/2,0,w)	$2U_{1}^{S}(2)⊕U_{2}^{S}(2) $
B:(u,v,0)	$3B_{1}^{S}(2) $
C:(u,u,w)	$3C_{1}^{S}(2) $
D:(u,0,w)	$2D_{1}^{S}(2)⊕D_{2}^{S}(2) $
E:(u,v,1/2)	$3E_{1}^{S}(2) $
GP:(u,v,w)	$3GP_{1}^{S}(2) $

12g

k-vector	A↑G(12)
A:(0,0,1/2)	$A_{1}^{S,+}(2)⊕A_{1}^{S,-}(2)⊕A_{3}^{S,+}(4)⊕A_{3}^{S,-}(4) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{3}^{S,+}(4)⊕Γ_{3}^{S,-}(4) $
H:(1/3,1/3,1/2)	$2H_{1}^{S}(2)⊕2H_{3}^{S}(4) $
K:(1/3,1/3,0)	$2K_{1}^{S}(2)⊕2K_{3}^{S}(4) $
L:(1/2,0,1/2)	$2L_{1}^{S,+}(2)⊕2L_{1}^{S,-}(2)⊕L_{2}^{S,+}(2)⊕L_{2}^{S,-}(2) $
M:(1/2,0,0)	$2M_{1}^{S,+}(2)⊕2M_{1}^{S,-}(2)⊕M_{2}^{S,+}(2)⊕M_{2}^{S,-}(2) $
Δ:(0,0,w)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2)⊕2Δ_{3}^{S}(4) $
P:(1/3,1/3,w)	$2P_{1}^{S}(2)⊕2P_{2}^{S}P_{3}^{S}(4) $
Λ:(u,u,0)	$4Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2) $
Σ:(u,0,0)	$3Σ_{1}^{S}(2)⊕3Σ_{2}^{S}(2) $
Q:(u,u,1/2)	$4Q_{1}^{S}(2)⊕2Q_{2}^{S}(2) $
R:(u,0,1/2)	$3R_{1}^{S}(2)⊕3R_{2}^{S}(2) $
U:(1/2,0,w)	$3U_{1}^{S}(2)⊕3U_{2}^{S}(2) $
B:(u,v,0)	$6B_{1}^{S}(2) $
C:(u,u,w)	$6C_{1}^{S}(2) $
D:(u,0,w)	$3D_{1}^{S}(2)⊕3D_{2}^{S}(2) $
E:(u,v,1/2)	$6E_{1}^{S}(2) $
GP:(u,v,w)	$6GP_{1}^{S}(2) $

12i

k-vector	A'↑G(12)
A:(0,0,1/2)	$A_{1}^{S,+}(2)⊕A_{2}^{S,-}(2)⊕A_{3}^{S,+}(4)⊕A_{3}^{S,-}(4) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{2}^{S,-}(2)⊕Γ_{3}^{S,+}(4)⊕Γ_{3}^{S,-}(4) $
H:(1/3,1/3,1/2)	$H_{1}^{S}(2)⊕H_{2}^{S}(2)⊕2H_{3}^{S}(4) $
K:(1/3,1/3,0)	$K_{1}^{S}(2)⊕K_{2}^{S}(2)⊕2K_{3}^{S}(4) $
L:(1/2,0,1/2)	$2L_{1}^{S,+}(2)⊕L_{1}^{S,-}(2)⊕L_{2}^{S,+}(2)⊕2L_{2}^{S,-}(2) $
M:(1/2,0,0)	$2M_{1}^{S,+}(2)⊕M_{1}^{S,-}(2)⊕M_{2}^{S,+}(2)⊕2M_{2}^{S,-}(2) $
Δ:(0,0,w)	$2Δ_{1}^{S}(2)⊕2Δ_{3}^{S}(4) $
P:(1/3,1/3,w)	$2P_{1}^{S}(2)⊕2P_{2}^{S}P_{3}^{S}(4) $
Λ:(u,u,0)	$3Λ_{1}^{S}(2)⊕3Λ_{2}^{S}(2) $
Σ:(u,0,0)	$4Σ_{1}^{S}(2)⊕2Σ_{2}^{S}(2) $
Q:(u,u,1/2)	$3Q_{1}^{S}(2)⊕3Q_{2}^{S}(2) $
R:(u,0,1/2)	$4R_{1}^{S}(2)⊕2R_{2}^{S}(2) $
U:(1/2,0,w)	$4U_{1}^{S}(2)⊕2U_{2}^{S}(2) $
B:(u,v,0)	$6B_{1}^{S}(2) $
C:(u,u,w)	$6C_{1}^{S}(2) $
D:(u,0,w)	$4D_{1}^{S}(2)⊕2D_{2}^{S}(2) $
E:(u,v,1/2)	$6E_{1}^{S}(2) $
GP:(u,v,w)	$6GP_{1}^{S}(2) $

24j

k-vector
$A↑G(24) $
$ $
A:(0,0,1/2)	$A_{1}^{S,+}(2)⊕A_{1}^{S,-}(2)⊕A_{2}^{S,+}(2)⊕A_{2}^{S,-}(2)⊕2A_{3}^{S,+}(4)⊕2A_{3}^{S,-}(4) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{2}^{S,-}(2)⊕2Γ_{3}^{S,+}(4)⊕2Γ_{3}^{S,-}(4) $
H:(1/3,1/3,1/2)	$2H_{1}^{S}(2)⊕2H_{2}^{S}(2)⊕4H_{3}^{S}(4) $
K:(1/3,1/3,0)	$2K_{1}^{S}(2)⊕2K_{2}^{S}(2)⊕4K_{3}^{S}(4) $
L:(1/2,0,1/2)	$3L_{1}^{S,+}(2)⊕3L_{1}^{S,-}(2)⊕3L_{2}^{S,+}(2)⊕3L_{2}^{S,-}(2) $
M:(1/2,0,0)	$3M_{1}^{S,+}(2)⊕3M_{1}^{S,-}(2)⊕3M_{2}^{S,+}(2)⊕3M_{2}^{S,-}(2) $
Δ:(0,0,w)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2)⊕4Δ_{3}^{S}(4) $
P:(1/3,1/3,w)	$4P_{1}^{S}(2)⊕4P_{2}^{S}P_{3}^{S}(4) $
Λ:(u,u,0)	$6Λ_{1}^{S}(2)⊕6Λ_{2}^{S}(2) $
Σ:(u,0,0)	$6Σ_{1}^{S}(2)⊕6Σ_{2}^{S}(2) $
Q:(u,u,1/2)	$6Q_{1}^{S}(2)⊕6Q_{2}^{S}(2) $
R:(u,0,1/2)	$6R_{1}^{S}(2)⊕6R_{2}^{S}(2) $
U:(1/2,0,w)	$6U_{1}^{S}(2)⊕6U_{2}^{S}(2) $
B:(u,v,0)	$12B_{1}^{S}(2) $
C:(u,u,w)	$12C_{1}^{S}(2) $
D:(u,0,w)	$6D_{1}^{S}(2)⊕6D_{2}^{S}(2) $
E:(u,v,1/2)	$12E_{1}^{S}(2) $