Information of spin space group $F\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:42.42.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,c |0,0,z)
8b	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,c |0,0,z)
8c	$\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(0,b,c |0,0,z)
8d	$.\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(a,0,c |0,0,z)
16e	$\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:(1,0,0)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Y:(0,1,0)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Z:(0,0,1)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
L:(1/2,1/2,1/2)	$\ce{^{1}{1}}\ce{^{\infty m}{1}} $
A:(u,0,1)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
B:(0,v,1)	$\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:(0,1,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}} $
E:(0,v,w)	$\ce{^{1}{m}}\ce{^{\infty}{1}} $
J:(u,0,w)	$\ce{^{1}{m}}\ce{^{\infty}{1}} $
M:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4a

k-vector	A1↑G(1)
Γ:(0,0,0)	$Γ_{1}^{S}(1) $
T:(1,0,0)	$T_{1}^{S}(1) $
Y:(0,1,0)	$Y_{1}^{S}(1) $
Z:(0,0,1)	$Z_{1}^{S}(1) $
L:(1/2,1/2,1/2)	$L_{1}^{S}(1) $
A:(u,0,1)	$A_{1}^{S}(1) $
B:(0,v,1)	$B_{1}^{S}(1) $
Δ:(0,v,0)	$Δ_{1}^{S}(1) $
H:(0,1,w)	$H_{1}^{S}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1) $
Σ:(u,0,0)	$Σ_{1}^{S}(1) $
E:(0,v,w)	$E_{1}^{S}(1) $
J:(u,0,w)	$J_{1}^{S}(1) $
M:(u,v,0)	$M_{1}^{S}(1) $
GP:(u,v,w)	$GP_{1}^{S}(1) $

8b

k-vector	A↑G(2)
Γ:(0,0,0)	$Γ_{1}^{S}(1)⊕Γ_{2}^{S}(1) $
T:(1,0,0)	$T_{3}^{S}(1)⊕T_{4}^{S}(1) $
Y:(0,1,0)	$Y_{3}^{S}(1)⊕Y_{4}^{S}(1) $
Z:(0,0,1)	$Z_{1}^{S}(1)⊕Z_{2}^{S}(1) $
L:(1/2,1/2,1/2)	$2L_{1}^{S}(1) $
A:(u,0,1)	$A_{1}^{S}(1)⊕A_{2}^{S}(1) $
B:(0,v,1)	$B_{1}^{S}(1)⊕B_{2}^{S}(1) $
Δ:(0,v,0)	$Δ_{1}^{S}(1)⊕Δ_{2}^{S}(1) $
H:(0,1,w)	$H_{3}^{S}(1)⊕H_{4}^{S}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1)⊕Λ_{2}^{S}(1) $
Σ:(u,0,0)	$Σ_{1}^{S}(1)⊕Σ_{2}^{S}(1) $
E:(0,v,w)	$E_{1}^{S}(1)⊕E_{2}^{S}(1) $
J:(u,0,w)	$J_{1}^{S}(1)⊕J_{2}^{S}(1) $
M:(u,v,0)	$2M_{1}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

8c

k-vector	A'↑G(2)
Γ:(0,0,0)	$Γ_{1}^{S}(1)⊕Γ_{3}^{S}(1) $
T:(1,0,0)	$T_{1}^{S}(1)⊕T_{3}^{S}(1) $
Y:(0,1,0)	$Y_{1}^{S}(1)⊕Y_{3}^{S}(1) $
Z:(0,0,1)	$Z_{1}^{S}(1)⊕Z_{3}^{S}(1) $
L:(1/2,1/2,1/2)	$2L_{1}^{S}(1) $
A:(u,0,1)	$A_{1}^{S}(1)⊕A_{2}^{S}(1) $
B:(0,v,1)	$2B_{1}^{S}(1) $
Δ:(0,v,0)	$2Δ_{1}^{S}(1) $
H:(0,1,w)	$H_{1}^{S}(1)⊕H_{3}^{S}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1)⊕Λ_{3}^{S}(1) $
Σ:(u,0,0)	$Σ_{1}^{S}(1)⊕Σ_{2}^{S}(1) $
E:(0,v,w)	$2E_{1}^{S}(1) $
J:(u,0,w)	$J_{1}^{S}(1)⊕J_{2}^{S}(1) $
M:(u,v,0)	$2M_{1}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

8d

k-vector	A'↑G(2)
Γ:(0,0,0)	$Γ_{1}^{S}(1)⊕Γ_{4}^{S}(1) $
T:(1,0,0)	$T_{1}^{S}(1)⊕T_{4}^{S}(1) $
Y:(0,1,0)	$Y_{1}^{S}(1)⊕Y_{4}^{S}(1) $
Z:(0,0,1)	$Z_{1}^{S}(1)⊕Z_{4}^{S}(1) $
L:(1/2,1/2,1/2)	$2L_{1}^{S}(1) $
A:(u,0,1)	$2A_{1}^{S}(1) $
B:(0,v,1)	$B_{1}^{S}(1)⊕B_{2}^{S}(1) $
Δ:(0,v,0)	$Δ_{1}^{S}(1)⊕Δ_{2}^{S}(1) $
H:(0,1,w)	$H_{1}^{S}(1)⊕H_{4}^{S}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1)⊕Λ_{4}^{S}(1) $
Σ:(u,0,0)	$2Σ_{1}^{S}(1) $
E:(0,v,w)	$E_{1}^{S}(1)⊕E_{2}^{S}(1) $
J:(u,0,w)	$2J_{1}^{S}(1) $
M:(u,v,0)	$2M_{1}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

16e

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:(1,0,0)	$T_{1}^{S}(1)⊕T_{2}^{S}(1)⊕T_{3}^{S}(1)⊕T_{4}^{S}(1) $
Y:(0,1,0)	$Y_{1}^{S}(1)⊕Y_{2}^{S}(1)⊕Y_{3}^{S}(1)⊕Y_{4}^{S}(1) $
Z:(0,0,1)	$Z_{1}^{S}(1)⊕Z_{2}^{S}(1)⊕Z_{3}^{S}(1)⊕Z_{4}^{S}(1) $
L:(1/2,1/2,1/2)	$4L_{1}^{S}(1) $
A:(u,0,1)	$2A_{1}^{S}(1)⊕2A_{2}^{S}(1) $
B:(0,v,1)	$2B_{1}^{S}(1)⊕2B_{2}^{S}(1) $
Δ:(0,v,0)	$2Δ_{1}^{S}(1)⊕2Δ_{2}^{S}(1) $
H:(0,1,w)	$H_{1}^{S}(1)⊕H_{2}^{S}(1)⊕H_{3}^{S}(1)⊕H_{4}^{S}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1)⊕Λ_{2}^{S}(1)⊕Λ_{3}^{S}(1)⊕Λ_{4}^{S}(1) $
Σ:(u,0,0)	$2Σ_{1}^{S}(1)⊕2Σ_{2}^{S}(1) $
E:(0,v,w)	$2E_{1}^{S}(1)⊕2E_{2}^{S}(1) $
J:(u,0,w)	$2J_{1}^{S}(1)⊕2J_{2}^{S}(1) $
M:(u,v,0)	$4M_{1}^{S}(1) $
GP:(u,v,w)	$4GP_{1}^{S}(1) $