Information of spin space group $P\ce{^{\text{-}1}{2_{1}}}\ce{^{\text{-}1}{2_{1}}}\ce{^{1}{2_{1}}}\ce{^{\infty m}{1}}$ ( nSSG:4.19.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\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{^{2}{2}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
B:(0,v,1/2)	$ \ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
C:(u,1/2,0)	$ \ce{^{2}{2}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
D:(1/2,v,0)	$ \ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$ \ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
E:(u,1/2,1/2)	$ \ce{^{2}{2}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
G:(1/2,0,w)	$ \ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
H:(0,1/2,w)	$ \ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
P:(1/2,v,1/2)	$ \ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
Q:(1/2,1/2,w)	$ \ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$ \ce{^{2}{2}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
K:(0,v,w)	$ \ce{^{-1}{2}}\ce{^{\infty}{1}} $
L:(1/2,v,w)	$ \ce{^{-1}{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{^{m}{2}}\ce{^{\infty}{1}} $
W:(u,v,1/2)	$ \ce{^{m}{2}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$ \ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4a

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