Information of spin space group $P\ce{^{\text{-}1}{\text{-}4}}\ce{^{\infty m}{1}}$ ( nSSG:3.81.1.1 )

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 ‖ 2001 | 0 }
b, -a, -c | -x, -y, -z	{ -1 ‖ -4+001 | 0 }
-b, a, -c | -x, -y, -z	{ -1 ‖ -4-001 | 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
b, -a, -c | x, -y, -z	{ 2 ‖ -4+001 | 0 }
-b, a, -c | x, -y, -z	{ 2 ‖ -4-001 | 0 }

Wyckoff Positions

WP	Site symmetry	Representative
1a	$\ce{^{-1}{-4}}..\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
1b	$\ce{^{-1}{-4}}..\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,0)
1c	$\ce{^{-1}{-4}}..\ce{^{\infty m}{1}} $	(1/2,1/2,0 | 0,0,0)
1d	$\ce{^{-1}{-4}}..\ce{^{\infty m}{1}} $	(1/2,1/2,1/2 | 0,0,0)
2e	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(0,0,c | 0,0,z)
2f	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(1/2,1/2,c | 0,0,z)
2g	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,z)
4h	$\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:(1/2,1/2,1/2)	$\ce{^{-1}{-4}}\ce{^{\infty m}{1}} $
Γ:(0,0,0)	$\ce{^{-1}{-4}}\ce{^{\infty m}{1}} $
M:(1/2,1/2,0)	$\ce{^{-1}{-4}}\ce{^{\infty m}{1}} $
Z:(0,0,1/2)	$\ce{^{-1}{-4}}\ce{^{\infty m}{1}} $
R:(0,1/2,1/2)	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $
X:(0,1/2,0)	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Λ:(0,0,w)	$\ce{^{-1}{-4}}\ce{^{\infty}{1}} $
V:(1/2,1/2,w)	$\ce{^{-1}{-4}}\ce{^{\infty}{1}} $
D:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
E:(u,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
S:(u,u,1/2)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
Σ:(u,u,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
T:(u,1/2,1/2)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
U:(0,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
W:(0,1/2,w)	$\ce{^{1}{2}}\ce{^{\infty}{1}} $
Y:(u,1/2,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
B:(0,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $
C:(u,u,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $
F:(u,1/2,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

2e

k-vector	k-vector-G↑	A↑G(2)
A:(1/2,1/2,-1/2)	E:(1/2,1/2,1/2)	$(A)E_{1}^{S}(2) $
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
M:(1/2,1/2,0)	A:(1/2,0,1/2)	$(M)A_{1}^{S}(2) $
R:(0,1/2,-1/2)	D:(0,1/2,1/2)	$2 (R)D_{1}^{S}(1)$	Spin Splitting
X:(0,1/2,0)	B:(0,0,1/2)	$2 (X)B_{1}^{S}(1)$	Spin Splitting
Z:(0,0,-1/2)	Z:(0,1/2,0)	$Z_{1}^{S}(2) $
B:(0,v,w)	GP:(0,-w,v)	$2(B)GP_{1}^{S}(1)$	Spin Splitting
C:(u,u,w)	GP:(u,-w,u)	$2(C)GP_{1}^{S}(1)$	Spin Splitting
D:(u,v,0)	F:(u,0,v)	$2(D)F_{1}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	F:(0,0,v)	$2(Δ)F_{1}^{S}(1)$	Spin Splitting
E:(u,v,-1/2)	G:(u,1/2,v)	$2(E)G_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	GP:(u,-w,1/2)	$2(F)GP_{1}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2) $
S:(u,u,-1/2)	G:(u,1/2,u)	$2(S)G_{1}^{S}(1)$	Spin Splitting
Σ:(u,u,0)	F:(u,0,u)	$2(Σ)F_{1}^{S}(1)$	Spin Splitting
T:(u,1/2,-1/2)	G:(u,1/2,1/2)	$2(T)G_{1}^{S}(1)$	Spin Splitting
U:(0,v,-1/2)	G:(0,1/2,v)	$2(U)G_{1}^{S}(1)$	Spin Splitting
V:(1/2,1/2,w)	U:(1/2,-w,1/2)	$(V)U_{1}^{S}(2) $
W:(0,1/2,w)	V:(0,-w,1/2)	$2 (W)V_{1}^{S}(1)$	Spin Splitting
Y:(u,1/2,0)	F:(u,0,1/2)	$2(Y)F_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,-w,v)	$2GP_{1}^{S}(1)$	Spin Splitting

2g

k-vector	k-vector-G↑	A↑G(2)
A:(1/2,1/2,-1/2)	E:(1/2,1/2,1/2)	$(A)E_{2}^{S}(2) $
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
M:(1/2,1/2,0)	A:(1/2,0,1/2)	$(M)A_{2}^{S}(2) $
R:(0,1/2,-1/2)	D:(0,1/2,1/2)	$2 (R)D_{1}^{S}(1)$	Spin Splitting
X:(0,1/2,0)	B:(0,0,1/2)	$2 (X)B_{1}^{S}(1)$	Spin Splitting
Z:(0,0,-1/2)	Z:(0,1/2,0)	$Z_{1}^{S}(2) $
B:(0,v,w)	GP:(0,-w,v)	$2(B)GP_{1}^{S}(1)$	Spin Splitting
C:(u,u,w)	GP:(u,-w,u)	$2(C)GP_{1}^{S}(1)$	Spin Splitting
D:(u,v,0)	F:(u,0,v)	$2(D)F_{1}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	F:(0,0,v)	$2(Δ)F_{1}^{S}(1)$	Spin Splitting
E:(u,v,-1/2)	G:(u,1/2,v)	$2(E)G_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	GP:(u,-w,1/2)	$2(F)GP_{1}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2) $
S:(u,u,-1/2)	G:(u,1/2,u)	$2(S)G_{1}^{S}(1)$	Spin Splitting
Σ:(u,u,0)	F:(u,0,u)	$2(Σ)F_{1}^{S}(1)$	Spin Splitting
T:(u,1/2,-1/2)	G:(u,1/2,1/2)	$2(T)G_{1}^{S}(1)$	Spin Splitting
U:(0,v,-1/2)	G:(0,1/2,v)	$2(U)G_{1}^{S}(1)$	Spin Splitting
V:(1/2,1/2,w)	U:(1/2,-w,1/2)	$(V)U_{2}^{S}(2) $
W:(0,1/2,w)	V:(0,-w,1/2)	$2 (W)V_{1}^{S}(1)$	Spin Splitting
Y:(u,1/2,0)	F:(u,0,1/2)	$2(Y)F_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,-w,v)	$2GP_{1}^{S}(1)$	Spin Splitting

2f

k-vector	k-vector-G↑	A↑G(2)
A:(1/2,1/2,-1/2)	E:(1/2,1/2,1/2)	$(A)E_{1}^{S}(2) $
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
M:(1/2,1/2,0)	A:(1/2,0,1/2)	$(M)A_{1}^{S}(2) $
R:(0,1/2,-1/2)	D:(0,1/2,1/2)	$2 (R)D_{2}^{S}(1)$	Spin Splitting
X:(0,1/2,0)	B:(0,0,1/2)	$2 (X)B_{2}^{S}(1)$	Spin Splitting
Z:(0,0,-1/2)	Z:(0,1/2,0)	$Z_{1}^{S}(2) $
B:(0,v,w)	GP:(0,-w,v)	$2(B)GP_{1}^{S}(1)$	Spin Splitting
C:(u,u,w)	GP:(u,-w,u)	$2(C)GP_{1}^{S}(1)$	Spin Splitting
D:(u,v,0)	F:(u,0,v)	$2(D)F_{1}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	F:(0,0,v)	$2(Δ)F_{1}^{S}(1)$	Spin Splitting
E:(u,v,-1/2)	G:(u,1/2,v)	$2(E)G_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	GP:(u,-w,1/2)	$2(F)GP_{1}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2) $
S:(u,u,-1/2)	G:(u,1/2,u)	$2(S)G_{1}^{S}(1)$	Spin Splitting
Σ:(u,u,0)	F:(u,0,u)	$2(Σ)F_{1}^{S}(1)$	Spin Splitting
T:(u,1/2,-1/2)	G:(u,1/2,1/2)	$2(T)G_{1}^{S}(1)$	Spin Splitting
U:(0,v,-1/2)	G:(0,1/2,v)	$2(U)G_{1}^{S}(1)$	Spin Splitting
V:(1/2,1/2,w)	U:(1/2,-w,1/2)	$(V)U_{1}^{S}(2) $
W:(0,1/2,w)	V:(0,-w,1/2)	$2 (W)V_{2}^{S}(1)$	Spin Splitting
Y:(u,1/2,0)	F:(u,0,1/2)	$2(Y)F_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,-w,v)	$2GP_{1}^{S}(1)$	Spin Splitting

4h

k-vector	k-vector-G↑	A↑G(4)
A:(1/2,1/2,-1/2)	E:(1/2,1/2,1/2)	$(A)E_{1}^{S}(2)⊕(A)E_{2}^{S}(2) $
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2) $
M:(1/2,1/2,0)	A:(1/2,0,1/2)	$(M)A_{1}^{S}(2)⊕(M)A_{2}^{S}(2) $
R:(0,1/2,-1/2)	D:(0,1/2,1/2)	$2(R)D_{1}^{S}(1)⊕2(R)D_{2}^{S}(1)$	Spin Splitting
X:(0,1/2,0)	B:(0,0,1/2)	$2(X)B_{1}^{S}(1)⊕2(X)B_{2}^{S}(1)$	Spin Splitting
Z:(0,0,-1/2)	Z:(0,1/2,0)	$Z_{1}^{S}(2)⊕Z_{2}^{S}(2) $
B:(0,v,w)	GP:(0,-w,v)	$4(B)GP_{1}^{S}(1)$	Spin Splitting
C:(u,u,w)	GP:(u,-w,u)	$4(C)GP_{1}^{S}(1)$	Spin Splitting
D:(u,v,0)	F:(u,0,v)	$4(D)F_{1}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	F:(0,0,v)	$4(Δ)F_{1}^{S}(1)$	Spin Splitting
E:(u,v,-1/2)	G:(u,1/2,v)	$4(E)G_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	GP:(u,-w,1/2)	$4(F)GP_{1}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
S:(u,u,-1/2)	G:(u,1/2,u)	$4(S)G_{1}^{S}(1)$	Spin Splitting
Σ:(u,u,0)	F:(u,0,u)	$4(Σ)F_{1}^{S}(1)$	Spin Splitting
T:(u,1/2,-1/2)	G:(u,1/2,1/2)	$4(T)G_{1}^{S}(1)$	Spin Splitting
U:(0,v,-1/2)	G:(0,1/2,v)	$4(U)G_{1}^{S}(1)$	Spin Splitting
V:(1/2,1/2,w)	U:(1/2,-w,1/2)	$(V)U_{1}^{S}(2)⊕(V)U_{2}^{S}(2) $
W:(0,1/2,w)	V:(0,-w,1/2)	$2(W)V_{1}^{S}(1)⊕2(W)V_{2}^{S}(1)$	Spin Splitting
Y:(u,1/2,0)	F:(u,0,1/2)	$4(Y)F_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,-w,v)	$4GP_{1}^{S}(1)$	Spin Splitting