Information of spin space group $C\ce{^{1}{2}}\ce{^{\text{-}1}{2}}\ce{^{\text{-}1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:5.21.1.1 )

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

Wyckoff Positions

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

Spin Brillouin Zone

Band Representations (Magnon)

4e

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
R:(1/2,-1/2,1/2)	L:(1/2,1/2,1/2)	$(R)L_{1}^{S}(2) $
S:(1/2,-1/2,0)	V:(1/2,1/2,0)	$(S)V_{1}^{S}(2) $
T:(1,0,1/2)	M:(0,1,1/2)	$(T)M_{1}^{S}(2) $
Y:(1,0,0)	Y:(0,1,0)	$Y_{1}^{S}(2) $
Z:(0,0,1/2)	A:(0,0,1/2)	$(Z)A_{1}^{S}(2) $
A:(u,0,1/2)	U:(0,u,1/2)	$(A)U_{1}^{S}(2) $
B:(0,v,1/2)	B:(-v,0,1/2)	$B_{1}^{S}(2) $
D:(1/2,1/2,w)	GP:(-1/2,1/2,w)	$(D)GP_{1}^{S}(2) $
Δ:(0,v,0)	B:(-v,0,0)	$(Δ)B_{1}^{S}(2) $
H:(0,-1,w)	B:(1,0,w)	$(H)B_{1}^{S}(2) $
Λ:(0,0,w)	B:(0,0,w)	$(Λ)B_{1}^{S}(2) $
Σ:(u,0,0)	Λ:(0,u,0)	$(Σ)Λ_{1}^{S}(2) $
K:(0,v,w)	B:(-v,0,w)	$2(K)B_{1}^{S}(1)$	Spin Splitting
M:(u,0,w)	GP:(0,u,w)	$(M)GP_{1}^{S}(2) $
P:(u,v,0)	GP:(-v,u,0)	$(P)GP_{1}^{S}(2) $
Q:(u,v,1/2)	GP:(-v,u,1/2)	$(Q)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(-v,u,w)	$2GP_{1}^{S}(1)$	Spin Splitting

4f

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
R:(1/2,-1/2,1/2)	L:(1/2,1/2,1/2)	$(R)L_{1}^{S}(2) $
S:(1/2,-1/2,0)	V:(1/2,1/2,0)	$(S)V_{1}^{S}(2) $
T:(1,0,1/2)	M:(0,1,1/2)	$(T)M_{2}^{S}(2) $
Y:(1,0,0)	Y:(0,1,0)	$Y_{1}^{S}(2) $
Z:(0,0,1/2)	A:(0,0,1/2)	$(Z)A_{2}^{S}(2) $
A:(u,0,1/2)	U:(0,u,1/2)	$(A)U_{2}^{S}(2) $
B:(0,v,1/2)	B:(-v,0,1/2)	$B_{1}^{S}(2) $
D:(1/2,1/2,w)	GP:(-1/2,1/2,w)	$(D)GP_{1}^{S}(2) $
Δ:(0,v,0)	B:(-v,0,0)	$(Δ)B_{1}^{S}(2) $
H:(0,-1,w)	B:(1,0,w)	$(H)B_{1}^{S}(2) $
Λ:(0,0,w)	B:(0,0,w)	$(Λ)B_{1}^{S}(2) $
Σ:(u,0,0)	Λ:(0,u,0)	$(Σ)Λ_{1}^{S}(2) $
K:(0,v,w)	B:(-v,0,w)	$2(K)B_{1}^{S}(1)$	Spin Splitting
M:(u,0,w)	GP:(0,u,w)	$(M)GP_{1}^{S}(2) $
P:(u,v,0)	GP:(-v,u,0)	$(P)GP_{1}^{S}(2) $
Q:(u,v,1/2)	GP:(-v,u,1/2)	$(Q)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(-v,u,w)	$2GP_{1}^{S}(1)$	Spin Splitting

8l

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2) $
R:(1/2,-1/2,1/2)	L:(1/2,1/2,1/2)	$2(R)L_{1}^{S}(2) $
S:(1/2,-1/2,0)	V:(1/2,1/2,0)	$2(S)V_{1}^{S}(2) $
T:(1,0,1/2)	M:(0,1,1/2)	$(T)M_{1}^{S}(2)⊕(T)M_{2}^{S}(2) $
Y:(1,0,0)	Y:(0,1,0)	$Y_{1}^{S}(2)⊕Y_{2}^{S}(2) $
Z:(0,0,1/2)	A:(0,0,1/2)	$(Z)A_{1}^{S}(2)⊕(Z)A_{2}^{S}(2) $
A:(u,0,1/2)	U:(0,u,1/2)	$(A)U_{1}^{S}(2)⊕(A)U_{2}^{S}(2) $
B:(0,v,1/2)	B:(-v,0,1/2)	$2B_{1}^{S}(2) $
D:(1/2,1/2,w)	GP:(-1/2,1/2,w)	$2(D)GP_{1}^{S}(2) $
Δ:(0,v,0)	B:(-v,0,0)	$2(Δ)B_{1}^{S}(2) $
H:(0,-1,w)	B:(1,0,w)	$2(H)B_{1}^{S}(2) $
Λ:(0,0,w)	B:(0,0,w)	$2(Λ)B_{1}^{S}(2) $
Σ:(u,0,0)	Λ:(0,u,0)	$(Σ)Λ_{1}^{S}(2)⊕(Σ)Λ_{2}^{S}(2) $
K:(0,v,w)	B:(-v,0,w)	$4(K)B_{1}^{S}(1)$	Spin Splitting
M:(u,0,w)	GP:(0,u,w)	$2(M)GP_{1}^{S}(2) $
P:(u,v,0)	GP:(-v,u,0)	$2(P)GP_{1}^{S}(2) $
Q:(u,v,1/2)	GP:(-v,u,1/2)	$2(Q)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(-v,u,w)	$4GP_{1}^{S}(1)$	Spin Splitting