Information of spin space group $P\ce{^{1}{b}}\ce{^{\text{-}1}{c}}\ce{^{1}{m}}\ce{^{\infty m}{1}}$ ( nSSG:26.57.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-a, b+1/2, -c+1/2 | x, y, z	{ 1 ‖ 2010 | 0 1/2 1/2 }
-a, b+1/2, c | x, y, z	{ 1 ‖ m100 | 0 1/2 0 }
a, b, -c+1/2 | x, y, z	{ 1 ‖ m001 | 0 0 1/2 }
a, -b+1/2, -c | -x, -y, -z	{ -1 ‖ 2100 | 0 1/2 0 }
-a, -b, c+1/2 | -x, -y, -z	{ -1 ‖ 2001 | 0 0 1/2 }
-a, -b, -c | -x, -y, -z	{ -1 ‖ -1 | 0 }
a, -b+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ m010 | 0 1/2 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, b+1/2, -c+1/2 | -x, y, z	{ m ‖ 2010 | 0 1/2 1/2 }
-a, b+1/2, c | -x, y, z	{ m ‖ m100 | 0 1/2 0 }
a, b, -c+1/2 | -x, y, z	{ m ‖ m001 | 0 0 1/2 }
a, -b+1/2, -c | x, -y, -z	{ 2 ‖ 2100 | 0 1/2 0 }
-a, -b, c+1/2 | x, -y, -z	{ 2 ‖ 2001 | 0 0 1/2 }
-a, -b, -c | x, -y, -z	{ 2 ‖ -1 | 0 }
a, -b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ m010 | 0 1/2 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{-1}{-1}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
4b	$\ce{^{-1}{-1}}\ce{^{\infty m}{1}} $	(1/2,0,0 | 0,0,0)
4c	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,1/4,0 | 0,0,0)
4d	$..\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(a,b,1/4 | 0,0,z)
8e	$\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}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
R:(1/2,1/2,1/2)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
S:(1/2,1/2,0)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
T:(0,1/2,1/2)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
U:(1/2,0,1/2)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
X:(1/2,0,0)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
Y:(0,1/2,0)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
Z:(0,0,1/2)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
A:(u,0,1/2)	$ \ce{^{m}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
B:(0,v,1/2)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
C:(u,1/2,0)	$ \ce{^{m}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
D:(1/2,v,0)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
E:(u,1/2,1/2)	$ \ce{^{m}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
G:(1/2,0,w)	$ \ce{^{1}{m}}\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
H:(0,1/2,w)	$ \ce{^{1}{m}}\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{1}{m}}\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
P:(1/2,v,1/2)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Q:(1/2,1/2,w)	$ \ce{^{1}{m}}\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$ \ce{^{m}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
K:(0,v,w)	$ \ce{^{-1}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
L:(1/2,v,w)	$ \ce{^{-1}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
M:(u,0,w)	$ \ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
N:(u,1/2,w)	$ \ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
V:(u,v,0)	$ \ce{^{-1}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
W:(u,v,1/2)	$ \ce{^{-1}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$ \ce{^{-1}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4d

k-vector	k-vector-G↑	A'↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{3}^{S}(2) $
R:(-1/2,1/2,-1/2)	R:(1/2,1/2,1/2)	$R_{1}^{S}R_{3}^{S}(4) $
S:(-1/2,1/2,0)	T:(0,1/2,1/2)	$(S)T_{1}^{S}T_{3}^{S}(4) $
T:(0,1/2,-1/2)	U:(1/2,0,1/2)	$(T)U_{1}^{S}U_{3}^{S}(4) $
U:(-1/2,0,-1/2)	S:(1/2,1/2,0)	$(U)S_{1}^{S}(2)⊕(U)S_{3}^{S}(2) $
X:(-1/2,0,0)	Y:(0,1/2,0)	$(X)Y_{1}^{S}(2)⊕(X)Y_{3}^{S}(2) $
Y:(0,1/2,0)	Z:(0,0,1/2)	$(Y)Z_{1}^{S}Z_{3}^{S}(4) $
Z:(0,0,-1/2)	X:(1/2,0,0)	$(Z)X_{1}^{S}(2)⊕(Z)X_{3}^{S}(2) $
A:(u,0,-1/2)	D:(1/2,-u,0)	$2(A)D_{1}^{S}(2) $
B:(0,v,-1/2)	G:(1/2,0,v)	$(B)G_{1}^{S}(2)⊕(B)G_{3}^{S}(2) $
C:(u,1/2,0)	B:(0,-u,1/2)	$(C)B_{1}^{S}B_{1}^{S}(4) $
D:(-1/2,v,0)	H:(0,1/2,v)	$(D)H_{1}^{S}(2)⊕(D)H_{3}^{S}(2) $
Δ:(0,v,0)	Λ:(0,0,v)	$(Δ)Λ_{1}^{S}(2)⊕(Δ)Λ_{3}^{S}(2) $
E:(u,1/2,-1/2)	P:(1/2,-u,1/2)	$(E)P_{1}^{S}P_{1}^{S}(4) $
G:(-1/2,0,w)	C:(-w,1/2,0)	$(G)C_{1}^{S}(2)⊕(G)C_{2}^{S}(2) $
H:(0,1/2,w)	A:(-w,0,1/2)	$(H)A_{1}^{S}A_{2}^{S}(4) $
Λ:(0,0,w)	Σ:(-w,0,0)	$(Λ)Σ_{1}^{S}(2)⊕(Λ)Σ_{2}^{S}(2) $
P:(-1/2,v,-1/2)	Q:(1/2,1/2,v)	$(P)Q_{1}^{S}(2)⊕(P)Q_{3}^{S}(2) $
Q:(-1/2,1/2,w)	E:(-w,1/2,1/2)	$(Q)E_{1}^{S}E_{2}^{S}(4) $
Σ:(u,0,0)	Δ:(0,-u,0)	$2(Σ)Δ_{1}^{S}(2) $
K:(0,v,w)	M:(-w,0,v)	$(K)M_{1}^{S}(2)⊕(K)M_{2}^{S}(2) $
L:(-1/2,v,w)	N:(-w,1/2,v)	$(L)N_{1}^{S}(2)⊕(L)N_{2}^{S}(2) $
M:(u,0,w)	V:(-w,-u,0)	$2(M)V_{1}^{S}(2) $
N:(u,1/2,w)	W:(-w,-u,1/2)	$(N)W_{1}^{S}W_{1}^{S}(4) $
V:(u,v,0)	K:(0,-u,v)	$2(V)K_{1}^{S}(2) $
W:(u,v,-1/2)	L:(1/2,-u,v)	$2(W)L_{1}^{S}(2) $
GP:(u,v,w)	GP:(-w,-u,v)	$2GP_{1}^{S}(2) $

8e

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) $
R:(-1/2,1/2,-1/2)	R:(1/2,1/2,1/2)	$R_{1}^{S}R_{3}^{S}(4)⊕R_{2}^{S}R_{4}^{S}(4) $
S:(-1/2,1/2,0)	T:(0,1/2,1/2)	$(S)T_{1}^{S}T_{3}^{S}(4)⊕(S)T_{2}^{S}T_{4}^{S}(4) $
T:(0,1/2,-1/2)	U:(1/2,0,1/2)	$(T)U_{1}^{S}U_{3}^{S}(4)⊕(T)U_{2}^{S}U_{4}^{S}(4) $
U:(-1/2,0,-1/2)	S:(1/2,1/2,0)	$(U)S_{1}^{S}(2)⊕(U)S_{2}^{S}(2)⊕(U)S_{3}^{S}(2)⊕(U)S_{4}^{S}(2) $
X:(-1/2,0,0)	Y:(0,1/2,0)	$(X)Y_{1}^{S}(2)⊕(X)Y_{2}^{S}(2)⊕(X)Y_{3}^{S}(2)⊕(X)Y_{4}^{S}(2) $
Y:(0,1/2,0)	Z:(0,0,1/2)	$(Y)Z_{1}^{S}Z_{3}^{S}(4)⊕(Y)Z_{2}^{S}Z_{4}^{S}(4) $
Z:(0,0,-1/2)	X:(1/2,0,0)	$(Z)X_{1}^{S}(2)⊕(Z)X_{2}^{S}(2)⊕(Z)X_{3}^{S}(2)⊕(Z)X_{4}^{S}(2) $
A:(u,0,-1/2)	D:(1/2,-u,0)	$2 (A)D_{1}^{S}(2)⊕2(A)D_{2}^{S}(2) $
B:(0,v,-1/2)	G:(1/2,0,v)	$(B)G_{1}^{S}(2)⊕(B)G_{2}^{S}(2)⊕(B)G_{3}^{S}(2)⊕(B)G_{4}^{S}(2) $
C:(u,1/2,0)	B:(0,-u,1/2)	$(C)B_{1}^{S}B_{1}^{S}(4)⊕(C)B_{2}^{S}B_{2}^{S}(4) $
D:(-1/2,v,0)	H:(0,1/2,v)	$(D)H_{1}^{S}(2)⊕(D)H_{2}^{S}(2)⊕(D)H_{3}^{S}(2)⊕(D)H_{4}^{S}(2) $
Δ:(0,v,0)	Λ:(0,0,v)	$(Δ)Λ_{1}^{S}(2)⊕(Δ)Λ_{2}^{S}(2)⊕(Δ)Λ_{3}^{S}(2)⊕(Δ)Λ_{4}^{S}(2) $
E:(u,1/2,-1/2)	P:(1/2,-u,1/2)	$(E)P_{1}^{S}P_{1}^{S}(4)⊕(E)P_{2}^{S}P_{2}^{S}(4) $
G:(-1/2,0,w)	C:(-w,1/2,0)	$2(G)C_{1}^{S}(2)⊕2(G)C_{2}^{S}(2) $
H:(0,1/2,w)	A:(-w,0,1/2)	$2(H)A_{1}^{S}A_{2}^{S}(4) $
Λ:(0,0,w)	Σ:(-w,0,0)	$2(Λ)Σ_{1}^{S}(2)⊕2(Λ)Σ_{2}^{S}(2) $
P:(-1/2,v,-1/2)	Q:(1/2,1/2,v)	$(P)Q_{1}^{S}(2)⊕(P)Q_{2}^{S}(2)⊕(P)Q_{3}^{S}(2)⊕(P)Q_{4}^{S}(2) $
Q:(-1/2,1/2,w)	E:(-w,1/2,1/2)	$2(Q)E_{1}^{S}E_{2}^{S}(4) $
Σ:(u,0,0)	Δ:(0,-u,0)	$2(Σ)Δ_{1}^{S}(2)⊕2(Σ)Δ_{2}^{S}(2) $
K:(0,v,w)	M:(-w,0,v)	$2(K)M_{1}^{S}(2)⊕2(K)M_{2}^{S}(2) $
L:(-1/2,v,w)	N:(-w,1/2,v)	$2(L)N_{1}^{S}(2)⊕2(L)N_{2}^{S}(2) $
M:(u,0,w)	V:(-w,-u,0)	$4(M)V_{1}^{S}(2) $
N:(u,1/2,w)	W:(-w,-u,1/2)	$2(N)W_{1}^{S}W_{1}^{S}(4) $
V:(u,v,0)	K:(0,-u,v)	$2(V)K_{1}^{S}(2)⊕2(V)K_{2}^{S}(2) $
W:(u,v,-1/2)	L:(1/2,-u,v)	$2(W)L_{1}^{S}(2)⊕2(W)L_{2}^{S}(2) $
GP:(u,v,w)	GP:(-w,-u,v)	$4GP_{1}^{S}(2) $