Information of spin space group $P\ce{^{1}{2_{1}}}\ce{^{\text{-}1}{2_{1}}}\ce{^{\text{-}1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:4.18.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 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, -b, c | -x, -y, -z	{ -1 ‖ 2001 | 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 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, -b, c | x, -y, -z	{ 2 ‖ 2001 | 0 }

Wyckoff Positions

WP	Site symmetry	Representative
2a	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,0,c | 0,0,0)
2b	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,0)
4c	$\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}} $
R:(1/2,1/2,1/2)	$ \ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $
S:(1/2,1/2,0)	$ \ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $
T:(0,1/2,1/2)	$ \ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $
U:(1/2,0,1/2)	$ \ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $
X:(1/2,0,0)	$ \ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $
Y:(0,1/2,0)	$ \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}} $
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}} $
C:(u,1/2,0)	$ \ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{\infty}{1}} $
D:(1/2,v,0)	$ \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}} $
E:(u,1/2,1/2)	$ \ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{\infty}{1}} $
G:(1/2,0,w)	$ \ce{^{m}{2}}\ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{\infty}{1}} $
H:(0,1/2,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}} $
P:(1/2,v,1/2)	$ \ce{^{m}{2}}\ce{^{2}{2}}\ce{^{-1}{2}}\ce{^{\infty}{1}} $
Q:(1/2,1/2,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}} $
K:(0,v,w)	$ \ce{^{m}{2}}\ce{^{\infty}{1}} $
L:(1/2,v,w)	$ \ce{^{m}{2}}\ce{^{\infty}{1}} $
M:(u,0,w)	$ \ce{^{-1}{2}}\ce{^{\infty}{1}} $
N:(u,1/2,w)	$ \ce{^{-1}{2}}\ce{^{\infty}{1}} $
V:(u,v,0)	$ \ce{^{-1}{2}}\ce{^{\infty}{1}} $
W:(u,v,1/2)	$ \ce{^{-1}{2}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$ \ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

mmmP

Band Representations (Magnon)

4c

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)	E:(1/2,1/2,1/2)	$(R)E_{1}^{S}E_{2}^{S}(4) $
S:(1/2,-1/2,0)	C:(1/2,1/2,0)	$(S)C_{1}^{S}C_{2}^{S}(4) $
T:(0,-1/2,1/2)	A:(1/2,0,1/2)	$(T)A_{1}^{S}(2)⊕(T)A_{2}^{S}(2) $
U:(1/2,0,1/2)	D:(0,1/2,1/2)	$(U)D_{1}^{S}D_{2}^{S}(4) $
X:(1/2,0,0)	Z:(0,1/2,0)	$(X)Z_{1}^{S}Z_{2}^{S}(4) $
Y:(0,-1/2,0)	Y:(1/2,0,0)	$Y_{1}^{S}(2)⊕Y_{2}^{S}(2) $
Z:(0,0,1/2)	B:(0,0,1/2)	$(Z)B_{1}^{S}(2)⊕(Z)B_{2}^{S}(2) $
A:(u,0,1/2)	V:(0,u,1/2)	$(A)V_{1}^{S}(2)⊕(A)V_{2}^{S}(2) $
B:(0,v,1/2)	F:(-v,0,1/2)	$2(B)F_{1}^{S}(2) $
C:(u,-1/2,0)	W:(1/2,u,0)	$(C)W_{1}^{S}(2)⊕(C)W_{2}^{S}(2) $
D:(1/2,v,0)	G:(-v,1/2,0)	$(D)G_{1}^{S}G_{1}^{S}(4) $
Δ:(0,v,0)	F:(-v,0,0)	$2(Δ)F_{1}^{S}(2) $
E:(u,-1/2,1/2)	U:(1/2,u,1/2)	$(E)U_{1}^{S}(1)⊕(E)U_{2}^{S}(2) $
G:(1/2,0,w)	G:(0,1/2,w)	$G_{1}^{S}G_{1}^{S}(4) $
H:(0,1/2,w)	F:(-1/2,0,w)	$2(H)F_{1}^{S}(2) $
Λ:(0,0,w)	F:(0,0,w)	$2(Λ)F_{1}^{S}(2) $
P:(1/2,v,1/2)	G:(-v,1/2,1/2)	$(P)G_{1}^{S}G_{1}^{S}(4) $
Q:(1/2,1/2,w)	G:(-1/2,1/2,w)	$(Q)G_{1}^{S}G_{1}^{S}(4) $
Σ:(u,0,0)	Λ:(0,u,0)	$(Σ)Λ_{1}^{S}(2)⊕(Σ)Λ_{2}^{S}(2) $
K:(0,v,w)	F:(-v,0,w)	$4(K)F_{1}^{S}(1)$	Spin Splitting
L:(1/2,v,w)	G:(-v,1/2,w)	$2(L)G_{1}^{S}G_{1}^{S}(2)$	Spin Splitting
M:(u,0,w)	GP:(0,u,w)	$2(M)GP_{1}^{S}(2) $
N:(u,1/2,w)	GP:(-1/2,u,w)	$2(N)GP_{1}^{S}(2) $
V:(u,v,0)	GP:(-v,u,0)	$2(V)GP_{1}^{S}(2) $
W:(u,v,1/2)	GP:(-v,u,1/2)	$2(W)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(-v,u,w)	$4GP_{1}^{S}(1)$	Spin Splitting