Information of spin space group $A\ce{^{1}{e}}\ce{^{1}{a}}\ce{^{1}{2}}\ce{^{\text{-}1}{(1/2 \ 0 \ 1/2)}}\ce{^{\infty m}{1}}$ ( nSSG:41.42.2.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,c | 0,0,0)
8b	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,c | 0,0,0)
8c	$\ce{^{-1}{m}}..\ce{^{\infty m}{1}} $	(0,b,c | 0,0,0)
8d	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,0,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
Γ:(0,0,0)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
T:(1,0,1/2)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Y:(1,0,0)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Z:(0,0,1/2)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
R:(1/2,1/2,1/2)	$\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
S:(1/2,1/2,0)	$\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
A:(u,0,1/2)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
B:(0,v,1/2)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Δ:(0,v,0)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
H:(1,0,w)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
Λ:(0,0,w)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
Σ:(u,0,0)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
D:(1/2,1/2,w)	$\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
K:(0,v,w)	$\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
M:(u,0,w)	$\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
P:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Q:(u,v,1/2)	$\ce{^{m}{2}}\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(8)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2)⊕Γ_{3}^{S}(2)⊕Γ_{4}^{S}(2) $
T:(1/2,0,-1)	$2T_{1}^{S}(4) $
Y:(0,0,-1)	$Y_{1}^{S}(2)⊕Y_{2}^{S}(2)⊕Y_{3}^{S}(2)⊕Y_{4}^{S}(2) $
Z:(1/2,0,0)	$2Z_{1}^{S}(4) $
R:(1/2,1/2,-1/2)	$2R_{1}^{S}R_{2}^{S}(4) $
S:(0,1/2,-1/2)	$2S_{1}^{S}S_{2}^{S}(4) $
A:(1/2,0,w)	$2A_{1}^{S}(4) $
B:(1/2,v,0)	$2B_{1}^{S}B_{2}^{S}(4) $
Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
H:(u,0,-1)	$2H_{1}^{S}(2)⊕2H_{2}^{S}(2) $
Λ:(u,0,0)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2) $
Σ:(0,0,w)	$Σ_{1}^{S}(2)⊕Σ_{2}^{S}(2)⊕Σ_{3}^{S}(2)⊕Σ_{4}^{S}(2) $
D:(u,1/2,-1/2)	$2D_{1}^{S}D_{1}^{S}(4) $
K:(u,v,0)	$4K_{1}^{S}(2) $
M:(u,0,w)	$2M_{1}^{S}(2)⊕2M_{2}^{S}(2) $
P:(0,v,w)	$2P_{1}^{S}(2)⊕2P_{2}^{S}(2) $
Q:(1/2,v,w)	$2Q_{1}^{S}(2)⊕2Q_{2}^{S}(2) $
GP:(u,v,w)	$4GP_{1}^{S}(2) $