Information of spin space group $C\ce{^{\text{-}1}{2/}}\ce{^{\text{-}1}{c}}\ce{^{\infty m}{1}}$ ( nSSG:2.15.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
4b	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,1/2,0 | 0,0,z)
4c	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(1/4,1/4,0 | 0,0,z)
4d	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(1/4,1/4,1/2 | 0,0,z)
4e	$\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,b,1/4 | 0,0,0)
8f	$\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:(0,0,1/2)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Γ:(0,0,0)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
M:(0,1,1/2)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Y:(0,1,0)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
L:(1/2,1/2,1/2)	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $
V:(1/2,1/2,0)	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $
B:(u,0,w)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Λ:(0,v,0)	$\ce{^{2}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
U:(0,v,1/2)	$\ce{^{2}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{m}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4a

k-vector	k-vector-G↑	Ag↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2) $
A:(0,0,1/2)	Z:(0,0,1/2)	$(A)Z_{1}^{S,+}(2) $
L:(1/2,1/2,1/2)	U:(1/2,0,1/2)	$2 (L)U_{1}^{S,+}(1)$	Spin Splitting
M:(0,1,1/2)	R:(1/2,1/2,1/2)	$(M)R_{1}^{S,+}(2) $
V:(1/2,1/2,0)	X:(1/2,0,0)	$2(V)X_{1}^{S,+}(1)$	Spin Splitting
Y:(0,1,0)	V:(1/2,1/2,0)	$(Y)V_{1}^{S,+}(2) $
Λ:(0,v,0)	GP:(v/2,v/2,0)	$(Λ)GP_{1}^{S}(2) $
U:(0,v,1/2)	GP:(v/2,v/2,1/2)	$(U)GP_{1}^{S}(2) $
B:(u,0,w)	GP:(u/2,-u/2,w)	$(B)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u/2+v/2,-u/2+v/2,w)	$2GP_{1}^{S}(1)$	Spin Splitting

4b

k-vector	k-vector-G↑	Ag↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2) $
A:(0,0,1/2)	Z:(0,0,1/2)	$(A)Z_{1}^{S,+}(2) $
L:(1/2,1/2,1/2)	U:(1/2,0,1/2)	$2 (L)U_{1}^{S,-}(1)$	Spin Splitting
M:(0,1,1/2)	R:(1/2,1/2,1/2)	$(M)R_{1}^{S,+}(2) $
V:(1/2,1/2,0)	X:(1/2,0,0)	$2(V)X_{1}^{S,-}(1)$	Spin Splitting
Y:(0,1,0)	V:(1/2,1/2,0)	$(Y)V_{1}^{S,+}(2) $
Λ:(0,v,0)	GP:(v/2,v/2,0)	$(Λ)GP_{1}^{S}(2) $
U:(0,v,1/2)	GP:(v/2,v/2,1/2)	$(U)GP_{1}^{S}(2) $
B:(u,0,w)	GP:(u/2,-u/2,w)	$(B)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u/2+v/2,-u/2+v/2,w)	$2GP_{1}^{S}(1)$	Spin Splitting

4c

k-vector	k-vector-G↑	Ag↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2) $
A:(0,0,1/2)	Z:(0,0,1/2)	$(A)Z_{1}^{S,+}(2) $
L:(1/2,1/2,1/2)	U:(1/2,0,1/2)	$2(L)U_{1}^{S,+}(1)$	Spin Splitting
M:(0,1,1/2)	R:(1/2,1/2,1/2)	$(M)R_{1}^{S,-}(2) $
V:(1/2,1/2,0)	X:(1/2,0,0)	$2 (V)X_{1}^{S,+}(1)$	Spin Splitting
Y:(0,1,0)	V:(1/2,1/2,0)	$(Y)V_{1}^{S,-}(2) $
Λ:(0,v,0)	GP:(v/2,v/2,0)	$(Λ)GP_{1}^{S}(2) $
U:(0,v,1/2)	GP:(v/2,v/2,1/2)	$(U)GP_{1}^{S}(2) $
B:(u,0,w)	GP:(u/2,-u/2,w)	$(B)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u/2+v/2,-u/2+v/2,w)	$2GP_{1}^{S}(1)$	Spin Splitting

4d

k-vector	k-vector-G↑	Ag↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2) $
A:(0,0,1/2)	Z:(0,0,1/2)	$(A)Z_{1}^{S,+}(2) $
L:(1/2,1/2,1/2)	U:(1/2,0,1/2)	$2(L)U_{1}^{S,-}(1)$	Spin Splitting
M:(0,1,1/2)	R:(1/2,1/2,1/2)	$(M)R_{1}^{S,-}(2) $
V:(1/2,1/2,0)	X:(1/2,0,0)	$2 (V)X_{1}^{S,-}(1)$	Spin Splitting
Y:(0,1,0)	V:(1/2,1/2,0)	$(Y)V_{1}^{S,-}(2) $
Λ:(0,v,0)	GP:(v/2,v/2,0)	$(Λ)GP_{1}^{S}(2) $
U:(0,v,1/2)	GP:(v/2,v/2,1/2)	$(U)GP_{1}^{S}(2) $
B:(u,0,w)	GP:(u/2,-u/2,w)	$(B)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u/2+v/2,-u/2+v/2,w)	$2GP_{1}^{S}(1)$	Spin Splitting

8f

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2) $
A:(0,0,1/2)	Z:(0,0,1/2)	$(A)Z_{1}^{S,+}(2)⊕(A)Z_{1}^{S,-}(2) $
L:(1/2,1/2,1/2)	U:(1/2,0,1/2)	$2 (L)U_{1}^{S,+}(1)⊕2 (L)U_{1}^{S,-}(1)$	Spin Splitting
M:(0,1,1/2)	R:(1/2,1/2,1/2)	$(M)R_{1}^{S,+}(2)⊕(M)R_{1}^{S,-}(2) $
V:(1/2,1/2,0)	X:(1/2,0,0)	$2(V)X_{1}^{S,+}(1)⊕2(V)X_{1}^{S,-}(1)$	Spin Splitting
Y:(0,1,0)	V:(1/2,1/2,0)	$(Y)V_{1}^{S,+}(2)⊕(Y)V_{1}^{S,-}(2) $
Λ:(0,v,0)	GP:(v/2,v/2,0)	$2(Λ)GP_{1}^{S}(2) $
U:(0,v,1/2)	GP:(v/2,v/2,1/2)	$2(U)GP_{1}^{S}(2) $
B:(u,0,w)	GP:(u/2,-u/2,w)	$2(B)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u/2+v/2,-u/2+v/2,w)	$4GP_{1}^{S}(1)$	Spin Splitting