Information of spin space group $P\ce{^{\text{-}1}{4_{2}/}}\ce{^{1}{m}}\ce{^{1}{n}}\ce{^{\text{-}1}{m}}\ce{^{\infty m}{1}}$ ( nSSG:58.136.1.1 )

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+1/2 | x, y, z	{ 1 ‖ 2100 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, -c+1/2 | x, y, z	{ 1 ‖ 2010 | 1/2 1/2 1/2 }
-a, -b, c | x, y, z	{ 1 ‖ 2001 | 0 }
-a, -b, -c | x, y, z	{ 1 ‖ -1 | 0 }
-a+1/2, b+1/2, c+1/2 | x, y, z	{ 1 ‖ m100 | 1/2 1/2 1/2 }
a+1/2, -b+1/2, c+1/2 | x, y, z	{ 1 ‖ m010 | 1/2 1/2 1/2 }
a, b, -c | x, y, z	{ 1 ‖ m001 | 0 }
-b+1/2, a+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ 4+001 | 1/2 1/2 1/2 }
b+1/2, -a+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ 4-001 | 1/2 1/2 1/2 }
b, a, -c | -x, -y, -z	{ -1 ‖ 2110 | 0 }
-b, -a, -c | -x, -y, -z	{ -1 ‖ 21-10 | 0 }
b+1/2, -a+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ -4+001 | 1/2 1/2 1/2 }
-b+1/2, a+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ -4-001 | 1/2 1/2 1/2 }
-b, -a, c | -x, -y, -z	{ -1 ‖ m110 | 0 }
b, a, c | -x, -y, -z	{ -1 ‖ m1-10 | 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
a+1/2, -b+1/2, -c+1/2 | -x, y, z	{ m ‖ 2100 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, -c+1/2 | -x, y, z	{ m ‖ 2010 | 1/2 1/2 1/2 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
-a, -b, -c | -x, y, z	{ m ‖ -1 | 0 }
-a+1/2, b+1/2, c+1/2 | -x, y, z	{ m ‖ m100 | 1/2 1/2 1/2 }
a+1/2, -b+1/2, c+1/2 | -x, y, z	{ m ‖ m010 | 1/2 1/2 1/2 }
a, b, -c | -x, y, z	{ m ‖ m001 | 0 }
-b+1/2, a+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 4+001 | 1/2 1/2 1/2 }
b+1/2, -a+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 4-001 | 1/2 1/2 1/2 }
b, a, -c | x, -y, -z	{ 2 ‖ 2110 | 0 }
-b, -a, -c | x, -y, -z	{ 2 ‖ 21-10 | 0 }
b+1/2, -a+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ -4+001 | 1/2 1/2 1/2 }
-b+1/2, a+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ -4-001 | 1/2 1/2 1/2 }
-b, -a, c | x, -y, -z	{ 2 ‖ m110 | 0 }
b, a, c | x, -y, -z	{ 2 ‖ m1-10 | 0 }

Wyckoff Positions

WP	Site symmetry	Representative
2a	$\ce{^{1}{m}}.\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
2b	$\ce{^{1}{m}}.\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,0)
4c	$\ce{^{1}{2}}/\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(0,1/2,0 | 0,0,z)
4d	$\ce{^{-1}{-4}}..\ce{^{\infty m}{1}} $	(0,1/2,1/4 | 0,0,0)
4e	$\ce{^{1}{2}}.\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(0,0,c | 0,0,0)
4f	$\ce{^{1}{m}}.\ce{^{-1}{2}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(a,a,0 | 0,0,0)
4g	$\ce{^{1}{m}}.\ce{^{-1}{m}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(a,-a,0 | 0,0,0)
8h	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,z)
8i	$\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(a,b,0 | 0,0,z)
8j	$..\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(a,a,c | 0,0,0)
16k	$\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}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Γ:(0,0,0)	$\ce{^{-1}{4/}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
M:(1/2,1/2,0)	$\ce{^{-1}{4/}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Z:(0,0,1/2)	$\ce{^{-1}{4/}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
R:(0,1/2,1/2)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
X:(0,1/2,0)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
Λ:(0,0,w)	$\ce{^{2}{4/}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
V:(1/2,1/2,w)	$\ce{^{2}{4/}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
S:(u,u,1/2)	$\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Σ:(u,u,0)	$\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
T:(u,1/2,1/2)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
U:(0,v,1/2)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
W:(0,1/2,w)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Y:(u,1/2,0)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
B:(0,v,w)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
C:(u,u,w)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
D:(u,v,0)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
E:(u,v,1/2)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
F:(u,1/2,w)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{m}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

4/mmmP

Band Representations (Magnon)

4c

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

8h

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

8i

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

16k

k-vector	k-vector-G↑	A↑G(16)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{2}^{S,-}(2)⊕Γ_{3}^{S,+}(2)⊕Γ_{3}^{S,-}(2)⊕Γ_{4}^{S,+}(2)⊕Γ_{4}^{S,-}(2) $
A:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$2 (A)R_{1}^{S}(4)⊕2 (A)R_{2}^{S}(4) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{1}^{S,+}S_{2}^{S,+}(4)⊕(M)S_{1}^{S,-}S_{2}^{S,-}(4)⊕(M)S_{3}^{S,+}S_{4}^{S,+}(4)⊕(M)S_{3}^{S,-}S_{4}^{S,-}(4) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$4 (R)T_{1}^{S,+}(2)⊕4 (R)T_{1}^{S,-}(2)$	Spin Splitting
X:(0,1/2,0)	Y:(0,1/2,0)	$4(X)Y_{1}^{S}(2)⊕4(X)Y_{2}^{S}(2)$	Spin Splitting
Z:(0,0,1/2)	Z:(0,0,1/2)	$2 Z_{1}^{S}(4)⊕2 Z_{2}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$4 Δ_{1}^{S}(1)⊕4 Δ_{2}^{S}(1)⊕4 Δ_{3}^{S}(1)⊕4 Δ_{4}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,0,w)	$2 Λ_{1}^{S}(2)⊕2 Λ_{2}^{S}(2)⊕2 Λ_{3}^{S}(2)⊕2 Λ_{4}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$4 (S)W_{1}^{S}(2)⊕4 (S)W_{2}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$4(Σ)V_{1}^{S}(2)⊕4(Σ)V_{2}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$8(T)E_{1}^{S}(2)$	Spin Splitting
U:(0,v,1/2)	B:(0,v,1/2)	$8(U)B_{1}^{S}(2)$	Spin Splitting
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(2 V)Q_{1}^{S}Q_{2}^{S}(4)⊕2 (V)Q_{3}^{S}Q_{4}^{S}(4) $
W:(0,1/2,w)	H:(0,1/2,w)	$8(W)H_{1}^{S}(2)$	Spin Splitting
Y:(u,1/2,0)	C:(u,1/2,0)	$4(Y)C_{1}^{S}C_{4}^{S}(2)⊕4(Y)C_{2}^{S}C_{3}^{S}(2)$	Spin Splitting
B:(0,v,w)	K:(0,v,w)	$8(B)K_{1}^{S}(1)⊕8(B)K_{2}^{S}(1)$	Spin Splitting
C:(u,u,w)	GP:(u,u,w)	$8(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$8(D)V_{1}^{S}(1)⊕8(D)V_{2}^{S}(1)$	Spin Splitting
E:(u,v,1/2)	W:(u,v,1/2)	$8(E)W_{1}^{S}(1)⊕8(E)W_{2}^{S}(1)$	Spin Splitting
F:(u,1/2,w)	N:(u,1/2,w)	$8(F)N_{1}^{S}N_{2}^{S}(2)$	Spin Splitting
GP:(u,v,w)	GP:(u,v,w)	$16GP_{1}^{S}(1)$	Spin Splitting