Information of spin space group $P\ce{^{\text{-}1}{2_{1}/}}\ce{^{\text{-}1}{m}}\ce{^{\infty m}{1}}$ ( nSSG:2.11.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
2a	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
2b	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(1/2,0,0 | 0,0,z)
2c	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,z)
2d	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(1/2,0,1/2 | 0,0,z)
2e	$\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(a,1/4,c | 0,0,0)
4f	$\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:(1/2,0,1/2)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
B:(0,0,1/2)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
C:(1/2,1/2,0)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
D:(0,1/2,1/2)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
E:(1/2,1/2,1/2)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Γ:(0,0,0)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Y:(1/2,0,0)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Z:(0,1/2,0)	$\ce{^{-1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
F:(u,0,w)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
G:(u,1/2,w)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Λ:(0,v,0)	$\ce{^{2}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
U:(1/2,v,1/2)	$\ce{^{2}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
V:(0,v,1/2)	$\ce{^{2}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
W:(1/2,v,0)	$\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)

2a

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

2b

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

2c

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

2d

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

4f

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:(1/2,0,1/2)	U:(1/2,0,1/2)	$(A)U_{1}^{S,+}(2)⊕(A)U_{1}^{S,-}(2) $
B:(0,0,1/2)	Z:(0,0,1/2)	$(B)Z_{1}^{S,+}(2)⊕(B)Z_{1}^{S,-}(2) $
C:(1/2,1/2,0)	V:(1/2,1/2,0)	$(C)V_{1}^{S,+}(2) ⊕ (C)V_{1}^{S,-}(2) $
D:(0,1/2,1/2)	T:(0,1/2,1/2)	$(D)T_{1}^{S,+}(2)⊕(D)T_{1}^{S,-}(2) $
E:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$(E)R_{1}^{S,+}(2)⊕(E)R_{1}^{S,-}(2) $
Y:(1/2,0,0)	X:(1/2,0,0)	$(Y)X_{1}^{S,+}(2)⊕(Y)X_{1}^{S,-}(2) $
Z:(0,1/2,0)	Y:(0,1/2,0)	$(Z)Y_{1}^{S,+}(2)⊕(Z)Y_{1}^{S,-}(2) $
Λ:(0,v,0)	GP:(0,v,0)	$2 (Λ)GP_{1}^{S}(2) $
U:(1/2,v,1/2)	GP:(1/2,v,1/2)	$2 (U)GP_{1}^{S}(2) $
V:(0,v,1/2)	GP:(0,v,1/2)	$2 (V)GP_{1}^{S}(2) $
W:(1/2,v,0)	GP:(1/2,v,0)	$2 (W)GP_{1}^{S}(2) $
F:(u,0,w)	GP:(u,0,w)	$2 (F)GP_{1}^{S}(2) $
G:(u,1/2,w)	GP:(u,1/2,w)	$2(G)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(1)$	Spin Splitting