Information of spin space group $I\ce{^{\text{-}1}{m}}\ce{^{\text{-}1}{m}}\ce{^{\text{-}1}{a}}\ce{^{\infty m}{1}}$ ( nSSG:24.74.1.1 )

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 ‖ 2100 | 0 }
-a, b+1/2, -c | x, y, z	{ 1 ‖ 2010 | 0 1/2 0 }
-a, -b+1/2, c | x, y, z	{ 1 ‖ 2001 | 0 1/2 0 }
-a, -b, -c | -x, -y, -z	{ -1 ‖ -1 | 0 }
-a, b, c | -x, -y, -z	{ -1 ‖ m100 | 0 }
a, -b+1/2, c | -x, -y, -z	{ -1 ‖ m010 | 0 1/2 0 }
a, b+1/2, -c | -x, -y, -z	{ -1 ‖ m001 | 0 1/2 0 }
a+1/2, b+1/2, c+1/2 | x, y, z	{ 1 ‖ 1 | 1/2 1/2 1/2 }
a+1/2, -b+1/2, -c+1/2 | x, y, z	{ 1 ‖ 2100 | 1/2 1/2 1/2 }
-a+1/2, b, -c+1/2 | x, y, z	{ 1 ‖ 2010 | 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+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ -1 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ m100 | 1/2 1/2 1/2 }
a+1/2, -b, c+1/2 | -x, -y, -z	{ -1 ‖ m010 | 1/2 0 1/2 }
a+1/2, b, -c+1/2 | -x, -y, -z	{ -1 ‖ m001 | 1/2 0 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
a, -b, -c | -x, y, z	{ m ‖ 2100 | 0 }
-a, b+1/2, -c | -x, y, z	{ m ‖ 2010 | 0 1/2 0 }
-a, -b+1/2, c | -x, y, z	{ m ‖ 2001 | 0 1/2 0 }
-a, -b, -c | x, -y, -z	{ 2 ‖ -1 | 0 }
-a, b, c | x, -y, -z	{ 2 ‖ m100 | 0 }
a, -b+1/2, c | x, -y, -z	{ 2 ‖ m010 | 0 1/2 0 }
a, b+1/2, -c | x, -y, -z	{ 2 ‖ m001 | 0 1/2 0 }
a+1/2, b+1/2, c+1/2 | -x, y, z	{ m ‖ 1 | 1/2 1/2 1/2 }
a+1/2, -b+1/2, -c+1/2 | -x, y, z	{ m ‖ 2100 | 1/2 1/2 1/2 }
-a+1/2, b, -c+1/2 | -x, y, z	{ m ‖ 2010 | 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+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ -1 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ m100 | 1/2 1/2 1/2 }
a+1/2, -b, c+1/2 | x, -y, -z	{ 2 ‖ m010 | 1/2 0 1/2 }
a+1/2, b, -c+1/2 | x, -y, -z	{ 2 ‖ m001 | 1/2 0 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{2}}/\ce{^{-1}{m}}..\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
4b	$\ce{^{1}{2}}/\ce{^{-1}{m}}..\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,0)
4c	$.\ce{^{1}{2}}/\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(1/4,1/4,1/4 | 0,0,0)
4d	$.\ce{^{1}{2}}/\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(1/4,1/4,3/4 | 0,0,0)
4e	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,c | 0,0,0)
8f	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,z)
8g	$.\ce{^{1}{2}}.\ce{^{\infty m}{1}} $	(1/4,b,1/4 | 0,0,z)
8h	$\ce{^{-1}{m}}..\ce{^{\infty m}{1}} $	(0,b,c | 0,0,0)
8i	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,1/4,c | 0,0,0)
16j	$\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}{m}}\ce{^{\infty m}{1}} $
X:(1,1,1)	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
R:(1/2,0,1/2)	$\ce{^{1}{2/}}\ce{^{2}{m}}\ce{^{\infty m}{1}} $
S:(0,1/2,1/2)	$\ce{^{1}{2/}}\ce{^{2}{m}}\ce{^{\infty m}{1}} $
T:(1/2,1/2,0)	$\ce{^{1}{2/}}\ce{^{2}{m}}\ce{^{\infty m}{1}} $
W:(1/2,1/2,1/2)	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$ \ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$\ce{^{2}{m}}\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
A:(0,v,w)	$\ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
B:(u,0,w)	$\ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
C:(u,v,0)	$\ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
D:(u,1/2,1/2)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
P:(1/2,1/2,w)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Q:(1/2,v,1/2)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{-1}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

8f

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{3}^{S}(2) $
R:(1/2,0,1/2)	R:(1/2,0,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
S:(0,1/2,1/2)	S:(0,1/2,1/2)	$S_{1}^{S}(2)⊕S_{2}^{S}(2) $
T:(1/2,1/2,0)	T:(1/2,1/2,0)	$T_{1}^{S}(2)⊕T_{2}^{S}(2) $
W:(1/2,1/2,1/2)	W:(1/2,1/2,1/2)	$W_{1}^{S}(4) $
X:(1,1,1)	X:(1,1,1)	$X_{1}^{S}(2)⊕X_{3}^{S}(2) $
D:(u,1/2,1/2)	D:(u,1/2,1/2)	$D_{1}^{S}(2)⊕D_{2}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
P:(1/2,1/2,w)	P:(1/2,1/2,w)	$P_{1}^{S}(2)⊕P_{2}^{S}(2) $
Q:(1/2,v,1/2)	Q:(1/2,v,1/2)	$Q_{1}^{S}(2)⊕Q_{2}^{S}(2) $
Σ:(u,0,0)	Σ:(u,0,0)	$2Σ_{1}^{S}(2) $
A:(0,v,w)	A:(0,v,w)	$2A_{1}^{S}(2) $
B:(u,0,w)	B:(u,0,w)	$2B_{1}^{S}(2) $
C:(u,v,0)	C:(u,v,0)	$2C_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$2GP_{1}^{S}(2) $

8g

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{3}^{S}(2) $
R:(1/2,0,1/2)	R:(1/2,0,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
S:(0,1/2,1/2)	S:(0,1/2,1/2)	$S_{1}^{S}(2)⊕S_{2}^{S}(2) $
T:(1/2,1/2,0)	T:(1/2,1/2,0)	$T_{1}^{S}(2)⊕T_{2}^{S}(2) $
W:(1/2,1/2,1/2)	W:(1/2,1/2,1/2)	$W_{1}^{S}(4) $
X:(1,1,1)	X:(1,1,1)	$X_{1}^{S}(2)⊕X_{3}^{S}(2) $
D:(u,1/2,1/2)	D:(u,1/2,1/2)	$D_{1}^{S}(2)⊕D_{2}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
P:(1/2,1/2,w)	P:(1/2,1/2,w)	$P_{1}^{S}(2)⊕P_{2}^{S}(2) $
Q:(1/2,v,1/2)	Q:(1/2,v,1/2)	$Q_{1}^{S}(2)⊕Q_{2}^{S}(2) $
Σ:(u,0,0)	Σ:(u,0,0)	$2Σ_{1}^{S}(2) $
A:(0,v,w)	A:(0,v,w)	$2A_{1}^{S}(2) $
B:(u,0,w)	B:(u,0,w)	$2B_{1}^{S}(2) $
C:(u,v,0)	C:(u,v,0)	$2C_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$2GP_{1}^{S}(2) $

16j

k-vector	k-vector-G↑	A↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2)⊕Γ_{3}^{S}(2)⊕Γ_{4}^{S}(2) $
R:(1/2,0,1/2)	R:(1/2,0,1/2)	$2 R_{1}^{S}(2)⊕2 R_{2}^{S}(2) $
S:(0,1/2,1/2)	S:(0,1/2,1/2)	$2 S_{1}^{S}(2)⊕2 S_{2}^{S}(2) $
T:(1/2,1/2,0)	T:(1/2,1/2,0)	$2 T_{1}^{S}(2)⊕2 T_{2}^{S}(2) $
W:(1/2,1/2,1/2)	W:(1/2,1/2,1/2)	$2 W_{1}^{S}(4) $
X:(1,1,1)	X:(1,1,1)	$X_{1}^{S}(2)⊕X_{2}^{S}(2)⊕X_{3}^{S}(2)⊕X_{4}^{S}(2) $
D:(u,1/2,1/2)	D:(u,1/2,1/2)	$2 D_{1}^{S}(2)⊕2 D_{2}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$2 Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$2 Λ_{1}^{S}(2)⊕2 Λ_{2}^{S}(2) $
P:(1/2,1/2,w)	P:(1/2,1/2,w)	$2 P_{1}^{S}(2)⊕2 P_{2}^{S}(2) $
Q:(1/2,v,1/2)	Q:(1/2,v,1/2)	$2 Q_{1}^{S}(2)⊕2 Q_{2}^{S}(2) $
Σ:(u,0,0)	Σ:(u,0,0)	$2 Σ_{1}^{S}(2)⊕2 Σ_{2}^{S}(2) $
A:(0,v,w)	A:(0,v,w)	$4A_{1}^{S}(2) $
B:(u,0,w)	B:(u,0,w)	$4B_{1}^{S}(2) $
C:(u,v,0)	C:(u,v,0)	$4C_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(2) $