Information of spin space group $P\ce{^{\text{-}1}{\text{-}4}}\ce{^{1}{m}}\ce{^{\text{-}1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:25.115.1.1 )

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 }
-a, b, c | x, y, z	{ 1 ‖ m100 | 0 }
a, -b, c | x, y, z	{ 1 ‖ m010 | 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	{ m ‖ 1 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
-a, b, c | -x, y, z	{ m ‖ m100 | 0 }
a, -b, c | -x, y, z	{ m ‖ m010 | 0 }
b, a, -c | x, -y, -z	{ 2 ‖ 2110 | 0 }
-b, -a, -c | x, -y, -z	{ 2 ‖ 21-10 | 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{^{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,z)
2f	$\ce{^{1}{2}}\ce{^{1}{m}}\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(1/2,1/2,c | 0,0,z)
2g	$\ce{^{1}{2}}\ce{^{1}{m}}\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,z)
4h	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(a,a,0 | 0,0,0)
4i	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(a,a,1/2 | 0,0,0)
4j	$.\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(a,0,c | 0,0,z)
4k	$.\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(a,1/2,c | 0,0,z)
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{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
X:(0,1/2,0)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Λ:(0,0,w)	$\ce{^{-1}{-4}}\ce{^{m}{2}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
V:(1/2,1/2,w)	$\ce{^{-1}{-4}}\ce{^{m}{2}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
S:(u,u,1/2)	$\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
Σ:(u,u,0)	$\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
T:(u,1/2,1/2)	$\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
U:(0,v,1/2)	$\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
W:(0,1/2,w)	$\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
Y:(u,1/2,0)	$\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
B:(0,v,w)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
C:(u,u,w)	$\ce{^{2}{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{^{m}{2}}\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↑	A1↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
A:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$(A)R_{1}^{S}(2) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{1}^{S}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$2(R)T_{1}^{S}(1)$	Spin Splitting
X:(0,1/2,0)	Y:(0,1/2,0)	$2(X)Y_{1}^{S}(1)$	Spin Splitting
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$(S)W_{1}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$(Σ)V_{1}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$2(T)E_{1}^{S}(1)$	Spin Splitting
U:(0,v,1/2)	B:(0,v,1/2)	$2(U)B_{1}^{S}(1)$	Spin Splitting
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(V)Q_{1}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$2(W)H_{1}^{S}(1)$	Spin Splitting
Y:(u,1/2,0)	C:(u,1/2,0)	$2(Y)C_{1}^{S}(1)$	Spin Splitting
B:(0,v,w)	K:(0,v,w)	$2(B)K_{1}^{S}(1)$	Spin Splitting
C:(u,u,w)	GP:(u,u,w)	$(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$4(D)V_{1}^{S}(1)$	Spin Splitting
E:(u,v,1/2)	W:(u,v,1/2)	$4(E)W_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	N:(u,1/2,w)	$4(F)N_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(1)$	Spin Splitting

2f

k-vector	k-vector-G↑	A1↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
A:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$(A)R_{2}^{S}(2) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{2}^{S}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$2(R)T_{3}^{S}(1)$	Spin Splitting
X:(0,1/2,0)	Y:(0,1/2,0)	$2(X)Y_{3}^{S}(1)$	Spin Splitting
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$(S)W_{1}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$(Σ)V_{1}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$2(T)E_{2}^{S}(1)$	Spin Splitting
U:(0,v,1/2)	B:(0,v,1/2)	$2(U)B_{1}^{S}(1)$	Spin Splitting
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(V)Q_{2}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$2(W)H_{3}^{S}(1)$	Spin Splitting
Y:(u,1/2,0)	C:(u,1/2,0)	$2(Y)C_{2}^{S}(1)$	Spin Splitting
B:(0,v,w)	K:(0,v,w)	$2(B)K_{1}^{S}(1)$	Spin Splitting
C:(u,u,w)	GP:(u,u,w)	$(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$4(D)V_{1}^{S}(1)$	Spin Splitting
E:(u,v,1/2)	W:(u,v,1/2)	$4(E)W_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	N:(u,1/2,w)	$4(F)N_{2}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(1)$	Spin Splitting

2g

k-vector	k-vector-G↑	A1↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
A:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$(A)R_{3}^{S}(2) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{3}^{S}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$2(R)T_{3}^{S}(1)$	Spin Splitting
X:(0,1/2,0)	Y:(0,1/2,0)	$2(X)Y_{3}^{S}(1)$	Spin Splitting
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$(S)W_{1}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$(Σ)V_{1}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$2(T)E_{2}^{S}(1)$	Spin Splitting
U:(0,v,1/2)	B:(0,v,1/2)	$2(U)B_{1}^{S}(1)$	Spin Splitting
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(V)Q_{3}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$2(W)H_{3}^{S}(1)$	Spin Splitting
Y:(u,1/2,0)	C:(u,1/2,0)	$2(Y)C_{2}^{S}(1)$	Spin Splitting
B:(0,v,w)	K:(0,v,w)	$2(B)K_{1}^{S}(1)$	Spin Splitting
C:(u,u,w)	GP:(u,u,w)	$(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$4(D)V_{1}^{S}(1)$	Spin Splitting
E:(u,v,1/2)	W:(u,v,1/2)	$4(E)W_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	N:(u,1/2,w)	$4(F)N_{2}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(1)$	Spin Splitting

4j

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)	R:(1/2,1/2,1/2)	$(A)R_{1}^{S}(2)⊕(A)R_{4}^{S}(2) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{1}^{S}(2)⊕(M)S_{4}^{S}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$2(R)T_{1}^{S}(1)⊕2(R)T_{4}^{S}(1)$	Spin Splitting
X:(0,1/2,0)	Y:(0,1/2,0)	$2(X)Y_{1}^{S}(1)⊕2(X)Y_{4}^{S}(1)$	Spin Splitting
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{4}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(1)⊕2Δ_{2}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{4}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$2(S)W_{1}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$2(Σ)V_{1}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$4(T)E_{1}^{S}(1)$	Spin Splitting
U:(0,v,1/2)	B:(0,v,1/2)	$2(U)B_{1}^{S}(1)⊕2(U)B_{2}^{S}(1)$	Spin Splitting
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(V)Q_{1}^{S}(2)⊕(V)Q_{4}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$2(W)H_{1}^{S}(1)⊕2(W)H_{4}^{S}(1)$	Spin Splitting
Y:(u,1/2,0)	C:(u,1/2,0)	$4(Y)C_{1}^{S}(1)$	Spin Splitting
B:(0,v,w)	K:(0,v,w)	$2(B)K_{1}^{S}(1)⊕2(B)K_{2}^{S}(1)$	Spin Splitting
C:(u,u,w)	GP:(u,u,w)	$2(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$4(D)V_{1}^{S}(1)$	Spin Splitting
E:(u,v,1/2)	W:(u,v,1/2)	$4(E)W_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	N:(u,1/2,w)	$4(F)N_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(1)$	Spin Splitting

4k

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)	R:(1/2,1/2,1/2)	$(A)R_{2}^{S}(2)⊕(A)R_{3}^{S}(2) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{2}^{S}(2)⊕(M)S_{3}^{S}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$2(R)T_{2}^{S}(1)⊕2(R)T_{3}^{S}(1)$	Spin Splitting
X:(0,1/2,0)	Y:(0,1/2,0)	$2(X)Y_{2}^{S}(1)⊕2(X)Y_{3}^{S}(1)$	Spin Splitting
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{4}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(1)⊕2Δ_{2}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{4}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$2(S)W_{1}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$2(Σ)V_{1}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$4(T)E_{2}^{S}(1)$	Spin Splitting
U:(0,v,1/2)	B:(0,v,1/2)	$2(U)B_{1}^{S}(1)⊕2(U)B_{2}^{S}(1)$	Spin Splitting
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(V)Q_{2}^{S}(2)⊕(V)Q_{3}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$2(W)H_{2}^{S}(1)⊕2(W)H_{3}^{S}(1)$	Spin Splitting
Y:(u,1/2,0)	C:(u,1/2,0)	$4(Y)C_{2}^{S}(1)$	Spin Splitting
B:(0,v,w)	K:(0,v,w)	$2(B)K_{1}^{S}(1)⊕2(B)K_{2}^{S}(1)$	Spin Splitting
C:(u,u,w)	GP:(u,u,w)	$2(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$4(D)V_{1}^{S}(1)$	Spin Splitting
E:(u,v,1/2)	W:(u,v,1/2)	$4(E)W_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	N:(u,1/2,w)	$4(F)N_{2}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(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)	R:(1/2,1/2,1/2)	$(A)R_{1}^{S}(2)⊕(A)R_{2}^{S}(2)⊕(A)R_{3}^{S}(2)⊕(A)R_{4}^{S}(2) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{1}^{S}(2)⊕(M)S_{2}^{S}(2)⊕(M)S_{3}^{S}(2)⊕(M)S_{4}^{S}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$2(R)T_{1}^{S}(1)⊕2(R)T_{2}^{S}(1)⊕2(R)T_{3}^{S}(1)⊕2(R)T_{4}^{S}(1)$	Spin Splitting
X:(0,1/2,0)	Y:(0,1/2,0)	$2(X)Y_{1}^{S}(1)⊕2(X)Y_{2}^{S}(1)⊕2(X)Y_{3}^{S}(1)⊕2(X)Y_{4}^{S}(1)$	Spin Splitting
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)	Δ:(0,v,0)	$4Δ_{1}^{S}(1)⊕4Δ_{2}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}(2)⊕Λ_{4}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$4(S)W_{1}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$4(Σ)V_{1}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$4(T)E_{1}^{S}(1)⊕4(T)E_{2}^{S}(1)$	Spin Splitting
U:(0,v,1/2)	B:(0,v,1/2)	$4(U)B_{1}^{S}(1)⊕4(U)B_{2}^{S}(1)$	Spin Splitting
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(V)Q_{1}^{S}(2)⊕(V)Q_{2}^{S}(2)⊕(V)Q_{3}^{S}(2)⊕(V)Q_{4}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$2(W)H_{1}^{S}(1)⊕2(W)H_{2}^{S}(1)⊕2(W)H_{3}^{S}(1)⊕2(W)H_{4}^{S}(1)$	Spin Splitting
Y:(u,1/2,0)	C:(u,1/2,0)	$4(Y)C_{1}^{S}(1)⊕4(Y)C_{2}^{S}(1)$	Spin Splitting
B:(0,v,w)	K:(0,v,w)	$4(B)K_{1}^{S}(1)⊕4(B)K_{2}^{S}(1)$	Spin Splitting
C:(u,u,w)	GP:(u,u,w)	$4(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$8(D)V_{1}^{S}(1)$	Spin Splitting
E:(u,v,1/2)	W:(u,v,1/2)	$8(E)W_{1}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	N:(u,1/2,w)	$4(F)N_{1}^{S}(1)⊕4(F)N_{2}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,v,w)	$8GP_{1}^{S}(1)$	Spin Splitting