Information of spin space group $P\ce{^{1}{4_{2}}}\ce{^{1}{b}}\ce{^{1}{c}}\ce{^{\text{-}1}{(1/2 \ 1/2 \ 0)}}\ce{^{\infty m}{1}}$ ( nSSG:106.101.2.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-b, a, c+1/2 | x, y, z	{ 1 ‖ 4+001 | 0 0 1/2 }
b, -a, c+1/2 | x, y, z	{ 1 ‖ 4-001 | 0 0 1/2 }
-a, -b, c | x, y, z	{ 1 ‖ 2001 | 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 }
-b+1/2, -a+1/2, c+1/2 | x, y, z	{ 1 ‖ m110 | 1/2 1/2 1/2 }
b+1/2, a+1/2, c+1/2 | x, y, z	{ 1 ‖ m1-10 | 1/2 1/2 1/2 }
a+1/2, b+1/2, c | -x, -y, -z	{ -1 ‖ 1 | 1/2 1/2 0 }
-b+1/2, a+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ 4+001 | 1/2 1/2 1/2 }
b+1/2, -a+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ 4-001 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, c | -x, -y, -z	{ -1 ‖ 2001 | 1/2 1/2 0 }
-a, b, c | -x, -y, -z	{ -1 ‖ m100 | 0 }
a, -b, c | -x, -y, -z	{ -1 ‖ m010 | 0 }
-b, -a, c+1/2 | -x, -y, -z	{ -1 ‖ m110 | 0 0 1/2 }
b, a, c+1/2 | -x, -y, -z	{ -1 ‖ m1-10 | 0 0 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-b, a, c+1/2 | -x, y, z	{ m ‖ 4+001 | 0 0 1/2 }
b, -a, c+1/2 | -x, y, z	{ m ‖ 4-001 | 0 0 1/2 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 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 }
-b+1/2, -a+1/2, c+1/2 | -x, y, z	{ m ‖ m110 | 1/2 1/2 1/2 }
b+1/2, a+1/2, c+1/2 | -x, y, z	{ m ‖ m1-10 | 1/2 1/2 1/2 }
a+1/2, b+1/2, c | x, -y, -z	{ 2 ‖ 1 | 1/2 1/2 0 }
-b+1/2, a+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 4+001 | 1/2 1/2 1/2 }
b+1/2, -a+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 4-001 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, c | x, -y, -z	{ 2 ‖ 2001 | 1/2 1/2 0 }
-a, b, c | x, -y, -z	{ 2 ‖ m100 | 0 }
a, -b, c | x, -y, -z	{ 2 ‖ m010 | 0 }
-b, -a, c+1/2 | x, -y, -z	{ 2 ‖ m110 | 0 0 1/2 }
b, a, c+1/2 | x, -y, -z	{ 2 ‖ m1-10 | 0 0 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{2}}\ce{^{-1}{m}}\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,0)
4b	$\ce{^{1}{2}}\ce{^{-1}{m}}\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,0,c | 0,0,0)
8c	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(3/4,3/4,c | 0,0,0)
8d	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,b,c | 0,0,0)
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
A:(1/2,1/2,1/2)	$\ce{^{1}{4}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
Γ:(0,0,0)	$\ce{^{1}{4}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
M:(1/2,1/2,0)	$\ce{^{1}{4}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
Z:(0,0,1/2)	$\ce{^{1}{4}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
R:(0,1/2,1/2)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
X:(0,1/2,0)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Λ:(0,0,w)	$\ce{^{1}{4}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
V:(1/2,1/2,w)	$\ce{^{1}{4}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
Δ:(0,v,0)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
S:(u,u,1/2)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Σ:(u,u,0)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
T:(u,1/2,1/2)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
U:(0,v,1/2)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
W:(0,1/2,w)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
Y:(u,1/2,0)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
B:(0,v,w)	$\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
C:(u,u,w)	$\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
D:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
E:(u,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
F:(u,1/2,w)	$\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty 2}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

16e

k-vector	A↑G(16)
A:(1/2,1/2,1/2)	$A_{1}^{S}A_{4}^{S}(4)⊕A_{2}^{S}A_{3}^{S}(4)⊕A_{5}^{S}A_{5}^{S}(8) $
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2)⊕Γ_{3}^{S}(2)⊕Γ_{4}^{S}(2)⊕2Γ_{5}^{S}(4) $
M:(1/2,1/2,0)	$M_{1}^{S}M_{3}^{S}(4)⊕M_{2}^{S}M_{4}^{S}(4)⊕2M_{5}^{S}(4) $
Z:(0,0,1/2)	$Z_{1}^{S}Z_{2}^{S}(4)⊕Z_{3}^{S}Z_{4}^{S}(4)⊕2Z_{5}^{S}(4) $
R:(0,1/2,1/2)	$4R_{1}^{S}(4) $
X:(0,1/2,0)	$4X_{1}^{S}(4) $
Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}(2)⊕Λ_{4}^{S}(2)⊕2Λ_{5}^{S}(4) $
V:(1/2,1/2,w)	$V_{1}^{S}(2)⊕V_{2}^{S}(2)⊕V_{3}^{S}(2)⊕V_{4}^{S}(2)⊕2V_{5}^{S}(4) $
Δ:(0,v,0)	$4Δ_{1}^{S}(2)⊕4Δ_{2}^{S}(2) $
S:(u,u,1/2)	$4S_{1}^{S}S_{2}^{S}(4) $
Σ:(u,u,0)	$4Σ_{1}^{S}(2)⊕4Σ_{2}^{S}(2) $
T:(u,1/2,1/2)	$4T_{1}^{S}T_{2}^{S}(4) $
U:(0,v,1/2)	$4U_{1}^{S}(2)⊕4U_{2}^{S}(2) $
W:(0,1/2,w)	$4W_{1}^{S}(4) $
Y:(u,1/2,0)	$4Y_{1}^{S}Y_{2}^{S}(4) $
B:(0,v,w)	$4B_{1}^{S}(2)⊕4B_{2}^{S}(2) $
C:(u,u,w)	$4C_{1}^{S}(2)⊕4C_{2}^{S}(2) $
D:(u,v,0)	$8D_{1}^{S}(2) $
E:(u,v,1/2)	$8E_{1}^{S}(2) $
F:(u,1/2,w)	$4F_{1}^{S}(2)⊕4F_{2}^{S}(2) $
GP:(u,v,w)	$8GP_{1}^{S}(2) $