Information of spin space group $P\ce{^{\text{-}1}{\text{-}4}}\ce{^{\text{-}1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:21.115.1.1 )

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 ‖ 2110 | 0 }
-b, -a, -c | x, y, z	{ 1 ‖ 21-10 | 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	{ -1 ‖ m100 | 0 }
a, -b, c | -x, -y, -z	{ -1 ‖ m010 | 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	{ m ‖ 2110 | 0 }
-b, -a, -c | -x, y, z	{ m ‖ 21-10 | 0 }
b, -a, -c | x, -y, -z	{ 2 ‖ -4+001 | 0 }
-b, a, -c | x, -y, -z	{ 2 ‖ -4-001 | 0 }
-a, b, c | x, -y, -z	{ 2 ‖ m100 | 0 }
a, -b, c | x, -y, -z	{ 2 ‖ m010 | 0 }

Wyckoff Positions

WP	Site symmetry	Representative
1a	$\ce{^{-1}{-4}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
1b	$\ce{^{-1}{-4}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/2,1/2,0 | 0,0,0)
1c	$\ce{^{-1}{-4}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/2,1/2,1/2 | 0,0,0)
1d	$\ce{^{-1}{-4}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,0)
2e	$\ce{^{1}{2}}\ce{^{-1}{m}}\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,0,c | 0,0,0)
2f	$\ce{^{1}{2}}\ce{^{-1}{m}}\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(1/2,1/2,c | 0,0,0)
2g	$\ce{^{1}{2}}\ce{^{-1}{m}}\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,0)
4h	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(a,a,0 | 0,0,z)
4i	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(a,a,1/2 | 0,0,z)
4j	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,0,c | 0,0,0)
4k	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,1/2,c | 0,0,0)
8l	$\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{^{-1}{2}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
Γ:(0,0,0)	$\ce{^{-1}{-4}}\ce{^{-1}{2}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
M:(1/2,1/2,0)	$\ce{^{-1}{-4}}\ce{^{-1}{2}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
Z:(0,0,1/2)	$\ce{^{-1}{-4}}\ce{^{-1}{2}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
R:(0,1/2,1/2)	$\ce{^{2}{2}}\ce{^{2}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
X:(0,1/2,0)	$\ce{^{2}{2}}\ce{^{2}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Λ:(0,0,w)	$\ce{^{-1}{-4}}\ce{^{-1}{2}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
V:(1/2,1/2,w)	$\ce{^{-1}{-4}}\ce{^{-1}{2}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
S:(u,u,1/2)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
Σ:(u,u,0)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
T:(u,1/2,1/2)	$\ce{^{2}{2}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
U:(0,v,1/2)	$\ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
W:(0,1/2,w)	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
Y:(u,1/2,0)	$\ce{^{2}{2}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
B:(0,v,w)	$\ce{^{-1}{2}}\ce{^{\infty}{1}} $
C:(u,u,w)	$\ce{^{1}{m}}\ce{^{\infty}{1}} $
D:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
E:(u,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
F:(u,1/2,w)	$\ce{^{-1}{2}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4h

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{4}^{S}(2) $
A:(1/2,1/2,1/2)	T:(1,0,1/2)	$(A)T_{1}^{S}(2)⊕(A)T_{4}^{S}(2) $
M:(1/2,1/2,0)	Y:(1,0,0)	$(M)Y_{1}^{S}(2)⊕(M)Y_{4}^{S}(2) $
R:(0,1/2,1/2)	R:(1/2,1/2,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
X:(0,1/2,0)	S:(1/2,1/2,0)	$(X)S_{1}^{S}(2)⊕(X)S_{2}^{S}(2) $
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{4}^{S}(2) $
Δ:(0,v,0)	P:(v,v,0)	$2(Δ)P_{1}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
S:(u,u,1/2)	A:(2*u,0,1/2)	$2(S)A_{1}^{S}(1)⊕2(S)A_{2}^{S}(1)$	Spin Splitting
Σ:(u,u,0)	Σ:(2*u,0,0)	$2Σ_{1}^{S}(1)⊕2Σ_{2}^{S}(1)$	Spin Splitting
T:(u,1/2,1/2)	Q:(1/2+u,1/2-u,1/2)	$2(T)Q_{1}^{S}(2) $
U:(0,v,1/2)	Q:(v,v,1/2)	$2(U)Q_{1}^{S}(2) $
V:(1/2,1/2,w)	H:(1,0,w)	$(V)H_{1}^{S}(2)⊕(V)H_{2}^{S}(2) $
W:(0,1/2,w)	D:(1/2,1/2,w)	$(W)D_{1}^{S}(2)⊕(W)D_{2}^{S}(2) $
Y:(u,1/2,0)	P:(1/2+u,1/2-u,0)	$2(Y)P_{1}^{S}(2) $
B:(0,v,w)	GP:(v,v,w)	$2(B)GP_{1}^{S}(2) $
C:(u,u,w)	M:(2*u,0,w)	$4(C)M_{1}^{S}(1)$	Spin Splitting
D:(u,v,0)	P:(u+v,-u+v,0)	$4(D)P_{1}^{S}(1)$	Spin Splitting
E:(u,v,1/2)	Q:(u+v,-u+v,1/2)	$4(E)Q_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	GP:(1/2+u,1/2-u,w)	$2(F)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u+v,-u+v,w)	$4GP_{1}^{S}(1)$	Spin Splitting

4i

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{4}^{S}(2) $
A:(1/2,1/2,1/2)	T:(1,0,1/2)	$(A)T_{2}^{S}(2)⊕(A)T_{3}^{S}(2) $
M:(1/2,1/2,0)	Y:(1,0,0)	$(M)Y_{1}^{S}(2)⊕(M)Y_{4}^{S}(2) $
R:(0,1/2,1/2)	R:(1/2,1/2,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
X:(0,1/2,0)	S:(1/2,1/2,0)	$(X)S_{1}^{S}(2)⊕(X)S_{2}^{S}(2) $
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{2}^{S}(2)⊕Z_{3}^{S}(2) $
Δ:(0,v,0)	P:(v,v,0)	$2(Δ)P_{1}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
S:(u,u,1/2)	A:(2*u,0,1/2)	$2(S)A_{1}^{S}(1)⊕2(S)A_{2}^{S}(1)$	Spin Splitting
Σ:(u,u,0)	Σ:(2*u,0,0)	$2Σ_{1}^{S}(1)⊕2Σ_{2}^{S}(1)$	Spin Splitting
T:(u,1/2,1/2)	Q:(1/2+u,1/2-u,1/2)	$2(T)Q_{1}^{S}(2) $
U:(0,v,1/2)	Q:(v,v,1/2)	$2(U)Q_{1}^{S}(2) $
V:(1/2,1/2,w)	H:(1,0,w)	$(V)H_{1}^{S}(2)⊕(V)H_{2}^{S}(2) $
W:(0,1/2,w)	D:(1/2,1/2,w)	$(W)D_{1}^{S}(2)⊕(W)D_{2}^{S}(2) $
Y:(u,1/2,0)	P:(1/2+u,1/2-u,0)	$2(Y)P_{1}^{S}(2) $
B:(0,v,w)	GP:(v,v,w)	$2(B)GP_{1}^{S}(2) $
C:(u,u,w)	M:(2*u,0,w)	$4(C)M_{1}^{S}(1)$	Spin Splitting
D:(u,v,0)	P:(u+v,-u+v,0)	$4(D)P_{1}^{S}(1)$	Spin Splitting
E:(u,v,1/2)	Q:(u+v,-u+v,1/2)	$4(E)Q_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	GP:(1/2+u,1/2-u,w)	$2(F)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u+v,-u+v,w)	$4GP_{1}^{S}(1)$	Spin Splitting

8l

k-vector	k-vector-G↑	A↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2)⊕Γ_{3}^{S}(2)⊕Γ_{4}^{S}(2) $
A:(1/2,1/2,1/2)	T:(1,0,1/2)	$(A)T_{1}^{S}(2)⊕(A)T_{2}^{S}(2)⊕(A)T_{3}^{S}(2)⊕(A)T_{4}^{S}(2) $
M:(1/2,1/2,0)	Y:(1,0,0)	$(M)Y_{1}^{S}(2)⊕(M)Y_{2}^{S}(2)⊕(M)Y_{3}^{S}(2)⊕(M)Y_{4}^{S}(2) $
R:(0,1/2,1/2)	R:(1/2,1/2,1/2)	$2R_{1}^{S}(2)⊕2R_{2}^{S}(2) $
X:(0,1/2,0)	S:(1/2,1/2,0)	$2(X)S_{1}^{S}(2)⊕2(X)S_{2}^{S}(2) $
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{2}^{S}(2)⊕Z_{3}^{S}(2)⊕Z_{4}^{S}(2) $
Δ:(0,v,0)	P:(v,v,0)	$4(Δ)P_{1}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2) $
S:(u,u,1/2)	A:(2*u,0,1/2)	$4(S)A_{1}^{S}(1)⊕4(S)A_{2}^{S}(1)$	Spin Splitting
Σ:(u,u,0)	Σ:(2*u,0,0)	$4Σ_{1}^{S}(1)⊕4Σ_{2}^{S}(1)$	Spin Splitting
T:(u,1/2,1/2)	Q:(1/2+u,1/2-u,1/2)	$4(T)Q_{1}^{S}(2) $
U:(0,v,1/2)	Q:(v,v,1/2)	$4(U)Q_{1}^{S}(2) $
V:(1/2,1/2,w)	H:(1,0,w)	$2(V)H_{1}^{S}(2)⊕2(V)H_{2}^{S}(2) $
W:(0,1/2,w)	D:(1/2,1/2,w)	$2(W)D_{1}^{S}(2)⊕2(W)D_{2}^{S}(2) $
Y:(u,1/2,0)	P:(1/2+u,1/2-u,0)	$4(Y)P_{1}^{S}(2) $
B:(0,v,w)	GP:(v,v,w)	$4(B)GP_{1}^{S}(2) $
C:(u,u,w)	M:(2*u,0,w)	$8(C)M_{1}^{S}(1)$	Spin Splitting
D:(u,v,0)	P:(u+v,-u+v,0)	$8(D)P_{1}^{S}(1)$	Spin Splitting
E:(u,v,1/2)	Q:(u+v,-u+v,1/2)	$8(E)Q_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	GP:(1/2+u,1/2-u,w)	$4(F)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u+v,-u+v,w)	$8GP_{1}^{S}(1)$	Spin Splitting