Information of spin space group $I\ce{^{1}{m}}\ce{^{\text{-}1}{m}}\ce{^{\text{-}1}{a}}\ce{^{\infty m}{1}}$ ( nSSG:12.74.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{2}}/\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
4b	$\ce{^{1}{2}}/\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,z)
4c	$.\ce{^{-1}{2}}/\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(1/4,1/4,1/4 | 0,0,0)
4d	$.\ce{^{-1}{2}}/\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(1/4,1/4,3/4 | 0,0,0)
4e	$\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,c | 0,0,0)
8f	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,z)
8g	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(1/4,b,1/4 | 0,0,0)
8h	$\ce{^{1}{m}}..\ce{^{\infty m}{1}} $	(0,b,c | 0,0,z)
8i	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,1/4,c | 0,0,0)
16j	$\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}} $
X:(1,1,1)	$\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
R:(1/2,0,1/2)	$\ce{^{2}{2/}}\ce{^{2}{m}}\ce{^{\infty m}{1}} $
S:(0,1/2,1/2)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
T:(1/2,1/2,0)	$\ce{^{2}{2/}}\ce{^{2}{m}}\ce{^{\infty m}{1}} $
W:(1/2,1/2,1/2)	$\ce{^{m}{m}}\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$ \ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{m}{m}}\ce{^{2}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
A:(0,v,w)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
B:(u,0,w)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
C:(u,v,0)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
D:(u,1/2,1/2)	$\ce{^{1}{2/}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
P:(1/2,1/2,w)	$\ce{^{2}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Q:(1/2,v,1/2)	$\ce{^{2}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{m}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4a

k-vector	k-vector-G↑	Ag↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2) $
R:(-1/2,1,1/2)	L:(1/2,1/2,1/2)	$(R)L_{1}^{S,+}(2) $
S:(0,1/2,1/2)	A:(0,0,1/2)	$2 (S)A_{1}^{S,+}(1)$	Spin Splitting
T:(-1/2,1/2,0)	V:(1/2,1/2,0)	$(T)V_{1}^{S,+}(2) $
W:(1/2,1/2,1/2)	U:(0,-1/2,1/2)	$(W)U_{1}^{S}(2) $
X:(-1,0,0)	Y:(0,1,0)	$(X)Y_{1}^{S,+}(2) $
D:(u,1/2,1/2)	U:(0,-u,1/2)	$2 (D)U_{1}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	B:(v,0,0)	$(Δ)B_{1}^{S}(2) $
Λ:(0,0,w)	B:(-w,0,w)	$(Λ)B_{1}^{S}(2) $
P:(1/2,1/2,w)	GP:(1/2-w,-1/2,w)	$(P)GP_{1}^{S}(2) $
Q:(1/2,v,1/2)	GP:(-1/2+v,-1/2,1/2)	$(Q)GP_{1}^{S}(2) $
Σ:(u,0,0)	Λ:(0,-u,0)	$(Σ)Λ_{1}^{S}(2) $
A:(0,v,w)	B:(v-w,0,w)	$2(A)B_{1}^{S}(1)$	Spin Splitting
B:(u,0,w)	GP:(-w,-u,w)	$(B)GP_{1}^{S}(2) $
C:(u,v,0)	GP:(v,-u,0)	$(C)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(v-w,-u,w)	$2GP_{1}^{S}(1)$	Spin Splitting

4b

k-vector	k-vector-G↑	Ag↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2) $
R:(-1/2,1,1/2)	L:(1/2,1/2,1/2)	$(R)L_{1}^{S,-}(2) $
S:(0,1/2,1/2)	A:(0,0,1/2)	$2 (S)A_{2}^{S,-}(1)$	Spin Splitting
T:(-1/2,1/2,0)	V:(1/2,1/2,0)	$(T)V_{1}^{S,+}(2) $
W:(1/2,1/2,1/2)	U:(0,-1/2,1/2)	$(W)U_{2}^{S}(2) $
X:(-1,0,0)	Y:(0,1,0)	$(X)Y_{1}^{S,+}(2) $
D:(u,1/2,1/2)	U:(0,-u,1/2)	$2 (D)U_{2}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	B:(v,0,0)	$(Δ)B_{1}^{S}(2) $
Λ:(0,0,w)	B:(-w,0,w)	$(Λ)B_{1}^{S}(2) $
P:(1/2,1/2,w)	GP:(1/2-w,-1/2,w)	$(P)GP_{1}^{S}(2) $
Q:(1/2,v,1/2)	GP:(-1/2+v,-1/2,1/2)	$(Q)GP_{1}^{S}(2) $
Σ:(u,0,0)	Λ:(0,-u,0)	$(Σ)Λ_{1}^{S}(2) $
A:(0,v,w)	B:(v-w,0,w)	$2(A)B_{1}^{S}(1)$	Spin Splitting
B:(u,0,w)	GP:(-w,-u,w)	$(B)GP_{1}^{S}(2) $
C:(u,v,0)	GP:(v,-u,0)	$(C)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(v-w,-u,w)	$2GP_{1}^{S}(1)$	Spin Splitting

8f

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2) $
R:(-1/2,1,1/2)	L:(1/2,1/2,1/2)	$(R)L_{1}^{S,+}(2)⊕(R)L_{1}^{S,-}(2) $
S:(0,1/2,1/2)	A:(0,0,1/2)	$2 (S)A_{1}^{S,+}(1)⊕2 (S)A_{1}^{S,-}(1)$	Spin Splitting
T:(-1/2,1/2,0)	V:(1/2,1/2,0)	$(T)V_{1}^{S,+}(2)⊕(T)V_{1}^{S,-}(2) $
W:(1/2,1/2,1/2)	U:(0,-1/2,1/2)	$2(W)U_{1}^{S}(2) $
X:(-1,0,0)	Y:(0,1,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{1}^{S,-}(2) $
D:(u,1/2,1/2)	U:(0,-u,1/2)	$4(D)U_{1}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	B:(v,0,0)	$(Δ)B_{1}^{S}(2)⊕(Δ)B_{2}^{S}(2) $
Λ:(0,0,w)	B:(-w,0,w)	$(Λ)B_{1}^{S}(2)⊕(Λ)B_{2}^{S}(2) $
P:(1/2,1/2,w)	GP:(1/2-w,-1/2,w)	$2(P)GP_{1}^{S}(2) $
Q:(1/2,v,1/2)	GP:(-1/2+v,-1/2,1/2)	$2(Q)GP_{1}^{S}(2) $
Σ:(u,0,0)	Λ:(0,-u,0)	$2(Σ)Λ_{1}^{S}(2) $
A:(0,v,w)	B:(v-w,0,w)	$2(A)B_{1}^{S}(1)⊕2(A)B_{2}^{S}(1)$	Spin Splitting
B:(u,0,w)	GP:(-w,-u,w)	$2(B)GP_{1}^{S}(2) $
C:(u,v,0)	GP:(v,-u,0)	$2(C)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(v-w,-u,w)	$4GP_{1}^{S}(1)$	Spin Splitting

8h

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:(-1/2,1,1/2)	L:(1/2,1/2,1/2)	$(R)L_{1}^{S,+}(2)⊕(R)L_{1}^{S,-}(2) $
S:(0,1/2,1/2)	A:(0,0,1/2)	$2 (S)A_{1}^{S,+}(1)⊕2 (S)A_{2}^{S,-}(1)$	Spin Splitting
T:(-1/2,1/2,0)	V:(1/2,1/2,0)	$(T)V_{1}^{S,+}(2)⊕(T)V_{1}^{S,-}(2) $
W:(1/2,1/2,1/2)	U:(0,-1/2,1/2)	$(W)U_{1}^{S}(2)⊕(W)U_{2}^{S}(2) $
X:(-1,0,0)	Y:(0,1,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{2}^{S,-}(2) $
D:(u,1/2,1/2)	U:(0,-u,1/2)	$2(D)U_{1}^{S}(1)⊕2(D)U_{2}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	B:(v,0,0)	$2 (Δ)B_{1}^{S}(2) $
Λ:(0,0,w)	B:(-w,0,w)	$2 (Λ)B_{1}^{S}(2) $
P:(1/2,1/2,w)	GP:(1/2-w,-1/2,w)	$2(P)GP_{1}^{S}(2) $
Q:(1/2,v,1/2)	GP:(-1/2+v,-1/2,1/2)	$2(Q)GP_{1}^{S}(2) $
Σ:(u,0,0)	Λ:(0,-u,0)	$(Σ)Λ_{1}^{S}(2)⊕(Σ)Λ_{2}^{S}(2) $
A:(0,v,w)	B:(v-w,0,w)	$4(A)B_{1}^{S}(1)$	Spin Splitting
B:(u,0,w)	GP:(-w,-u,w)	$2(B)GP_{1}^{S}(2) $
C:(u,v,0)	GP:(v,-u,0)	$2(C)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(v-w,-u,w)	$4GP_{1}^{S}(1)$	Spin Splitting

16j

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) $
R:(-1/2,1,1/2)	L:(1/2,1/2,1/2)	$2(R)L_{1}^{S,+}(2)⊕2(R)L_{1}^{S,-}(2) $
S:(0,1/2,1/2)	A:(0,0,1/2)	$2(S)A_{1}^{S,+}(1)⊕2(S)A_{1}^{S,-}(1)⊕2(S)A_{2}^{S,+}(1)⊕2(S)A_{2}^{S,-}(1)$	Spin Splitting
T:(-1/2,1/2,0)	V:(1/2,1/2,0)	$2(T)V_{1}^{S,+}(2)⊕2(T)V_{1}^{S,-}(2) $
W:(1/2,1/2,1/2)	U:(0,-1/2,1/2)	$2(W)U_{1}^{S}(2)⊕2(W)U_{2}^{S}(2) $
X:(-1,0,0)	Y:(0,1,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{1}^{S,-}(2)⊕(X)Y_{2}^{S,+}(2)⊕(X)Y_{2}^{S,-}(2) $
D:(u,1/2,1/2)	U:(0,-u,1/2)	$4(D)U_{1}^{S}(1)⊕4(D)U_{2}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	B:(v,0,0)	$2(Δ)B_{1}^{S}(2)⊕2(Δ)B_{2}^{S}(2) $
Λ:(0,0,w)	B:(-w,0,w)	$2(Λ)B_{1}^{S}(2)⊕2(Λ)B_{2}^{S}(2) $
P:(1/2,1/2,w)	GP:(1/2-w,-1/2,w)	$4(P)GP_{1}^{S}(2) $
Q:(1/2,v,1/2)	GP:(-1/2+v,-1/2,1/2)	$4(Q)GP_{1}^{S}(2) $
Σ:(u,0,0)	Λ:(0,-u,0)	$2(Σ)Λ_{1}^{S}(2)⊕2(Σ)Λ_{2}^{S}(2) $
A:(0,v,w)	B:(v-w,0,w)	$4(A)B_{1}^{S}(1)⊕4(A)B_{2}^{S}(1)$	Spin Splitting
B:(u,0,w)	GP:(-w,-u,w)	$4(B)GP_{1}^{S}(2) $
C:(u,v,0)	GP:(v,-u,0)	$4(C)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(v-w,-u,w)	$8GP_{1}^{S}(1)$	Spin Splitting