Information of spin space group $P\ce{^{1}{m}}\ce{^{\text{-}1}{n}}\ce{^{1}{a}}\ce{^{\infty m}{1}}$ ( nSSG:28.53.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
2a	$\ce{^{-1}{2}}/\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
2b	$\ce{^{-1}{2}}/\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(1/2,0,0 | 0,0,0)
2c	$\ce{^{-1}{2}}/\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(1/2,1/2,0 | 0,0,0)
2d	$\ce{^{-1}{2}}/\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(0,1/2,0 | 0,0,0)
4e	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,0)
4f	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,1/2,0 | 0,0,0)
4g	$.\ce{^{1}{2}}.\ce{^{\infty m}{1}} $	(1/4,b,1/4 | 0,0,z)
4h	$\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(0,b,c | 0,0,z)
8i	$\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)

4g

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

4h

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

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