Information of spin space group $I\ce{^{1}{4_{1}/}}\ce{^{1}{a}}\ce{^{\infty m}{1}}$ ( nSSG:88.88.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-b+3/4, a+1/4, c+1/4 | x, y, z	{ 1 ‖ 4+001 | 3/4 1/4 1/4 }
b+1/4, -a+1/4, c+1/4 | x, y, z	{ 1 ‖ 4-001 | 1/4 1/4 1/4 }
-a, -b+1/2, c | x, y, z	{ 1 ‖ 2001 | 0 1/2 0 }
-a, -b, -c | x, y, z	{ 1 ‖ -1 | 0 }
b+1/4, -a+3/4, -c+3/4 | x, y, z	{ 1 ‖ -4+001 | 1/4 3/4 3/4 }
-b+3/4, a+3/4, -c+3/4 | x, y, z	{ 1 ‖ -4-001 | 3/4 3/4 3/4 }
a, b+1/2, -c | x, y, z	{ 1 ‖ m001 | 0 1/2 0 }
a+1/2, b+1/2, c+1/2 | x, y, z	{ 1 ‖ 1 | 1/2 1/2 1/2 }
-b+1/4, a+3/4, c+3/4 | x, y, z	{ 1 ‖ 4+001 | 1/4 3/4 3/4 }
b+3/4, -a+3/4, c+3/4 | x, y, z	{ 1 ‖ 4-001 | 3/4 3/4 3/4 }
-a+1/2, -b, c+1/2 | x, y, z	{ 1 ‖ 2001 | 1/2 0 1/2 }
-a+1/2, -b+1/2, -c+1/2 | x, y, z	{ 1 ‖ -1 | 1/2 1/2 1/2 }
b+3/4, -a+1/4, -c+1/4 | x, y, z	{ 1 ‖ -4+001 | 3/4 1/4 1/4 }
-b+1/4, a+1/4, -c+1/4 | x, y, z	{ 1 ‖ -4-001 | 1/4 1/4 1/4 }
a+1/2, b, -c+1/2 | x, y, z	{ 1 ‖ m001 | 1/2 0 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-b+3/4, a+1/4, c+1/4 | -x, y, z	{ m ‖ 4+001 | 3/4 1/4 1/4 }
b+1/4, -a+1/4, c+1/4 | -x, y, z	{ m ‖ 4-001 | 1/4 1/4 1/4 }
-a, -b+1/2, c | -x, y, z	{ m ‖ 2001 | 0 1/2 0 }
-a, -b, -c | -x, y, z	{ m ‖ -1 | 0 }
b+1/4, -a+3/4, -c+3/4 | -x, y, z	{ m ‖ -4+001 | 1/4 3/4 3/4 }
-b+3/4, a+3/4, -c+3/4 | -x, y, z	{ m ‖ -4-001 | 3/4 3/4 3/4 }
a, b+1/2, -c | -x, y, z	{ m ‖ m001 | 0 1/2 0 }
a+1/2, b+1/2, c+1/2 | -x, y, z	{ m ‖ 1 | 1/2 1/2 1/2 }
-b+1/4, a+3/4, c+3/4 | -x, y, z	{ m ‖ 4+001 | 1/4 3/4 3/4 }
b+3/4, -a+3/4, c+3/4 | -x, y, z	{ m ‖ 4-001 | 3/4 3/4 3/4 }
-a+1/2, -b, c+1/2 | -x, y, z	{ m ‖ 2001 | 1/2 0 1/2 }
-a+1/2, -b+1/2, -c+1/2 | -x, y, z	{ m ‖ -1 | 1/2 1/2 1/2 }
b+3/4, -a+1/4, -c+1/4 | -x, y, z	{ m ‖ -4+001 | 3/4 1/4 1/4 }
-b+1/4, a+1/4, -c+1/4 | -x, y, z	{ m ‖ -4-001 | 1/4 1/4 1/4 }
a+1/2, b, -c+1/2 | -x, y, z	{ m ‖ m001 | 1/2 0 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{-4}}..\ce{^{\infty m}{1}} $	(0,1/4,1/8 |0,0,z)
4b	$\ce{^{1}{-4}}..\ce{^{\infty m}{1}} $	(0,1/4,5/8 |0,0,z)
8c	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,0 |0,0,z)
8d	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,1/2 |0,0,z)
8e	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(0,1/4,c |0,0,z)
16f	$\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}{4/}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
M:(1,1,1)	$\ce{^{1}{4/}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
P:(1/2,1/2,1/2)	$\ce{^{m}{4/}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
X:(1/2,1/2,0)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
N:(1/2,0,1/2)	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $
Λ:(0,0,w)	$\ce{^{1}{4/}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
W:(1/2,1/2,w)	$\ce{^{1}{2/}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Q:(1/2,v,1/2)	$\ce{^{m}{-1}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Y:(u,1-u,0)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Δ:(u,u,0)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
A:(u,u,w)	$\ce{^{m}{-1}}\ce{^{\infty}{1}} $
B:(u,0,w)	$\ce{^{m}{-1}}\ce{^{\infty}{1}} $
C:(u,v,0)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{m}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

16f

k-vector	A↑G(8)
Γ:(0,0,0)	$Γ_{1}^{S,+}(1)⊕Γ_{1}^{S,-}(1)⊕Γ_{2}^{S,+}(1)⊕Γ_{2}^{S,-}(1)⊕Γ_{3}^{S,+}Γ_{4}^{S,+}(2)⊕Γ_{3}^{S,-}Γ_{4}^{S,-}(2) $
M:(1,1,1)	$2M_{1}^{S}(2)⊕2M_{2}^{S}(2) $
P:(1/2,1/2,1/2)	$2P_{1}^{S}P_{4}^{S}(2)⊕2P_{2}^{S}P_{3}^{S}(2) $
X:(1/2,1/2,0)	$4X_{1}^{S}(2) $
N:(1/2,0,1/2)	$4N_{1}^{S,+}(1)⊕4N_{1}^{S,-}(1) $
Λ:(0,0,w)	$2Λ_{1}^{S}(1)⊕2Λ_{2}^{S}(1)⊕2Λ_{3}^{S}Λ_{4}^{S}(2) $
W:(1/2,1/2,w)	$4W_{1}^{S}W_{2}^{S}(2) $
Q:(1/2,v,1/2)	$8Q_{1}^{S}(1) $
Σ:(u,0,0)	$4Σ_{1}^{S}(1)⊕4Σ_{2}^{S}(1) $
Y:(u,1-u,0)	$4Y_{1}^{S}(1)⊕4Y_{2}^{S}(1) $
Δ:(u,u,0)	$4Δ_{1}^{S}(1)⊕4Δ_{2}^{S}(1) $
A:(u,u,w)	$8A_{1}^{S}(1) $
B:(u,0,w)	$8B_{1}^{S}(1) $
C:(u,v,0)	$4C_{1}^{S}(1)⊕4C_{2}^{S}(1) $
GP:(u,v,w)	$8GP_{1}^{S}(1) $

4a

k-vector	A↑G(2)
Γ:(0,0,0)	$Γ_{1}^{S,+}(1)⊕Γ_{2}^{S,-}(1) $
M:(1,1,1)	$M_{1}^{S}(2) $
P:(1/2,1/2,1/2)	$P_{1}^{S}P_{4}^{S}(2) $
X:(1/2,1/2,0)	$X_{1}^{S}(2) $
N:(1/2,0,1/2)	$N_{1}^{S,+}(1)⊕N_{1}^{S,-}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1)⊕Λ_{2}^{S}(1) $
W:(1/2,1/2,w)	$W_{1}^{S}W_{2}^{S}(2) $
Q:(1/2,v,1/2)	$2Q_{1}^{S}(1) $
Σ:(u,0,0)	$Σ_{1}^{S}(1)⊕Σ_{2}^{S}(1) $
Y:(u,1-u,0)	$Y_{1}^{S}(1)⊕Y_{2}^{S}(1) $
Δ:(u,u,0)	$Δ_{1}^{S}(1)⊕Δ_{2}^{S}(1) $
A:(u,u,w)	$2A_{1}^{S}(1) $
B:(u,0,w)	$2B_{1}^{S}(1) $
C:(u,v,0)	$C_{1}^{S}(1)⊕C_{2}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

4b

k-vector	A↑G(2)
Γ:(0,0,0)	$Γ_{1}^{S,+}(1)⊕Γ_{2}^{S,-}(1) $
M:(1,1,1)	$M_{1}^{S}(2) $
P:(1/2,1/2,1/2)	$P_{2}^{S}P_{3}^{S}(2) $
X:(1/2,1/2,0)	$X_{1}^{S}(2) $
N:(1/2,0,1/2)	$N_{1}^{S,+}(1)⊕N_{1}^{S,-}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1)⊕Λ_{2}^{S}(1) $
W:(1/2,1/2,w)	$W_{1}^{S}W_{2}^{S}(2) $
Q:(1/2,v,1/2)	$2Q_{1}^{S}(1) $
Σ:(u,0,0)	$Σ_{1}^{S}(1)⊕Σ_{2}^{S}(1) $
Y:(u,1-u,0)	$Y_{1}^{S}(1)⊕Y_{2}^{S}(1) $
Δ:(u,u,0)	$Δ_{1}^{S}(1)⊕Δ_{2}^{S}(1) $
A:(u,u,w)	$2A_{1}^{S}(1) $
B:(u,0,w)	$2B_{1}^{S}(1) $
C:(u,v,0)	$C_{1}^{S}(1)⊕C_{2}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

8c

k-vector	Ag↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S,+}(1)⊕Γ_{2}^{S,+}(1)⊕Γ_{3}^{S,+}Γ_{4}^{S,+}(2) $
M:(1,1,1)	$M_{1}^{S}(2)⊕M_{2}^{S}(2) $
P:(1/2,1/2,1/2)	$P_{1}^{S}P_{4}^{S}(2)⊕P_{2}^{S}P_{3}^{S}(2) $
X:(1/2,1/2,0)	$2X_{1}^{S}(2) $
N:(1/2,0,1/2)	$3N_{1}^{S,+}(1)⊕N_{1}^{S,-}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1)⊕Λ_{2}^{S}(1)⊕Λ_{3}^{S}Λ_{4}^{S}(2) $
W:(1/2,1/2,w)	$2W_{1}^{S}W_{2}^{S}(2) $
Q:(1/2,v,1/2)	$4Q_{1}^{S}(1) $
Σ:(u,0,0)	$2Σ_{1}^{S}(1)⊕2Σ_{2}^{S}(1) $
Y:(u,1-u,0)	$2Y_{1}^{S}(1)⊕2Y_{2}^{S}(1) $
Δ:(u,u,0)	$2Δ_{1}^{S}(1)⊕2Δ_{2}^{S}(1) $
A:(u,u,w)	$4A_{1}^{S}(1) $
B:(u,0,w)	$4B_{1}^{S}(1) $
C:(u,v,0)	$2C_{1}^{S}(1)⊕2C_{2}^{S}(1) $
GP:(u,v,w)	$4GP_{1}^{S}(1) $

8d

k-vector	Ag↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S,+}(1)⊕Γ_{2}^{S,+}(1)⊕Γ_{3}^{S,+}Γ_{4}^{S,+}(2) $
M:(1,1,1)	$M_{1}^{S}(2)⊕M_{2}^{S}(2) $
P:(1/2,1/2,1/2)	$P_{1}^{S}P_{4}^{S}(2)⊕P_{2}^{S}P_{3}^{S}(2) $
X:(1/2,1/2,0)	$2X_{1}^{S}(2) $
N:(1/2,0,1/2)	$N_{1}^{S,+}(1)⊕3N_{1}^{S,-}(1) $
Λ:(0,0,w)	$Λ_{1}^{S}(1)⊕Λ_{2}^{S}(1)⊕Λ_{3}^{S}Λ_{4}^{S}(2) $
W:(1/2,1/2,w)	$2W_{1}^{S}W_{2}^{S}(2) $
Q:(1/2,v,1/2)	$4Q_{1}^{S}(1) $
Σ:(u,0,0)	$2Σ_{1}^{S}(1)⊕2Σ_{2}^{S}(1) $
Y:(u,1-u,0)	$2Y_{1}^{S}(1)⊕2Y_{2}^{S}(1) $
Δ:(u,u,0)	$2Δ_{1}^{S}(1)⊕2Δ_{2}^{S}(1) $
A:(u,u,w)	$4A_{1}^{S}(1) $
B:(u,0,w)	$4B_{1}^{S}(1) $
C:(u,v,0)	$2C_{1}^{S}(1)⊕2C_{2}^{S}(1) $
GP:(u,v,w)	$4GP_{1}^{S}(1) $

8e

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S,+}(1)⊕Γ_{1}^{S,-}(1)⊕Γ_{2}^{S,+}(1)⊕Γ_{2}^{S,-}(1) $
M:(1,1,1)	$2M_{1}^{S}(2) $
P:(1/2,1/2,1/2)	$P_{1}^{S}P_{4}^{S}(2)⊕P_{2}^{S}P_{3}^{S}(2) $
X:(1/2,1/2,0)	$2X_{1}^{S}(2) $
N:(1/2,0,1/2)	$2N_{1}^{S,+}(1)⊕2N_{1}^{S,-}(1) $
Λ:(0,0,w)	$2Λ_{1}^{S}(1)⊕2Λ_{2}^{S}(1) $
W:(1/2,1/2,w)	$2W_{1}^{S}W_{2}^{S}(2) $
Q:(1/2,v,1/2)	$4Q_{1}^{S}(1) $
Σ:(u,0,0)	$2Σ_{1}^{S}(1)⊕2Σ_{2}^{S}(1) $
Y:(u,1-u,0)	$2Y_{1}^{S}(1)⊕2Y_{2}^{S}(1) $
Δ:(u,u,0)	$2Δ_{1}^{S}(1)⊕2Δ_{2}^{S}(1) $
A:(u,u,w)	$4A_{1}^{S}(1) $
B:(u,0,w)	$4B_{1}^{S}(1) $
C:(u,v,0)	$2C_{1}^{S}(1)⊕2C_{2}^{S}(1) $
GP:(u,v,w)	$4GP_{1}^{S}(1) $