Information of spin space group $P\ce{^{\text{-}1}{m}}\ce{^{\text{-}1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}}$ ( nSSG:10.47.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
1a	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
1b	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(1/2,0,0 | 0,0,0)
1c	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,0)
1d	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(1/2,0,1/2 | 0,0,0)
1e	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(0,1/2,0 | 0,0,0)
1f	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(1/2,1/2,0 | 0,0,0)
1g	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(0,1/2,1/2 | 0,0,0)
1h	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(1/2,1/2,1/2 | 0,0,0)
2i	$\ce{^{-1}{2}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,0)
2j	$\ce{^{-1}{2}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(a,0,1/2 | 0,0,0)
2k	$\ce{^{-1}{2}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(a,1/2,0 | 0,0,0)
2l	$\ce{^{-1}{2}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(a,1/2,1/2 | 0,0,0)
2m	$\ce{^{-1}{m}}\ce{^{-1}{2}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(0,b,0 | 0,0,0)
2n	$\ce{^{-1}{m}}\ce{^{-1}{2}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(0,b,1/2 | 0,0,0)
2o	$\ce{^{-1}{m}}\ce{^{-1}{2}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(1/2,b,0 | 0,0,0)
2p	$\ce{^{-1}{m}}\ce{^{-1}{2}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(1/2,b,1/2 | 0,0,0)
2q	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,c | 0,0,0)
2r	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,0)
2s	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/2,0,c | 0,0,0)
2t	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/2,1/2,c | 0,0,0)
4u	$\ce{^{-1}{m}}..\ce{^{\infty m}{1}} $	(0,b,c | 0,0,0)
4v	$\ce{^{-1}{m}}..\ce{^{\infty m}{1}} $	(1/2,b,c | 0,0,0)
4w	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,0,c | 0,0,0)
4x	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,1/2,c | 0,0,0)
4y	$..\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(a,b,0 | 0,0,z)
4z	$..\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(a,b,1/2 | 0,0,z)
8alpha	$\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{^{-1}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
B:(0,v,1/2)	$ \ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
C:(u,1/2,0)	$ \ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
D:(1/2,v,0)	$ \ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$ \ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
E:(u,1/2,1/2)	$ \ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
G:(1/2,0,w)	$ \ce{^{2}{m}}\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
H:(0,1/2,w)	$ \ce{^{2}{m}}\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{2}{m}}\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
P:(1/2,v,1/2)	$ \ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Q:(1/2,1/2,w)	$ \ce{^{2}{m}}\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$ \ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
K:(0,v,w)	$ \ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
L:(1/2,v,w)	$ \ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
M:(u,0,w)	$ \ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
N:(u,1/2,w)	$ \ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
V:(u,v,0)	$ \ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
W:(u,v,1/2)	$ \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)

4y

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)	E:(1/2,1/2,1/2)	$(R)E_{1}^{S,+}(2)⊕(R)E_{2}^{S,-}(2) $
S:(1/2,1/2,0)	A:(1/2,0,1/2)	$(S)A_{1}^{S,+}(2)⊕(S)A_{2}^{S,-}(2) $
T:(0,1/2,-1/2)	D:(0,1/2,1/2)	$(T)D_{1}^{S,+}(2)⊕(T)D_{2}^{S,-}(2) $
U:(1/2,0,-1/2)	C:(1/2,1/2,0)	$(U)C_{1}^{S,+}(2)⊕(U)C_{2}^{S,-}(2) $
X:(1/2,0,0)	Y:(1/2,0,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{2}^{S,-}(2) $
Y:(0,1/2,0)	B:(0,0,1/2)	$(Y)B_{1}^{S,+}(2)⊕(Y)B_{2}^{S,-}(2) $
Z:(0,0,-1/2)	Z:(0,1/2,0)	$Z_{1}^{S,+}(2)⊕Z_{2}^{S,-}(2) $
A:(u,0,-1/2)	G:(u,1/2,0)	$2(A)G_{1}^{S}(2) $
B:(0,v,-1/2)	G:(0,1/2,v)	$2(B)G_{1}^{S}(2) $
C:(u,1/2,0)	F:(u,0,1/2)	$2(C)F_{1}^{S}(2) $
D:(1/2,v,0)	F:(1/2,0,v)	$2(D)F_{1}^{S}(2) $
Δ:(0,v,0)	F:(0,0,v)	$2(Δ)F_{1}^{S}(2) $
E:(u,1/2,-1/2)	G:(u,1/2,1/2)	$2(E)G_{1}^{S}(2) $
G:(1/2,0,w)	W:(1/2,-w,0)	$(G)W_{1}^{S}(2)⊕(G)W_{2}^{S}(2) $
H:(0,1/2,w)	V:(0,-w,1/2)	$(H)V_{1}^{S}(2)⊕(H)V_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
P:(1/2,v,-1/2)	G:(1/2,1/2,v)	$2(P)G_{1}^{S}(2) $
Q:(1/2,1/2,w)	U:(1/2,-w,1/2)	$(Q)U_{1}^{S}(2)⊕(Q)U_{2}^{S}(2) $
Σ:(u,0,0)	F:(u,0,0)	$2(Σ)F_{1}^{S}(2) $
K:(0,v,w)	GP:(0,-w,v)	$2(K)GP_{1}^{S}(2) $
L:(1/2,v,w)	GP:(1/2,-w,v)	$2(L)GP_{1}^{S}(2) $
M:(u,0,w)	GP:(u,-w,0)	$2(M)GP_{1}^{S}(2) $
N:(u,1/2,w)	GP:(u,-w,1/2)	$2(N)GP_{1}^{S}(2) $
V:(u,v,0)	F:(u,0,v)	$4(V)F_{1}^{S}(1)$	Spin Splitting
W:(u,v,-1/2)	G:(u,1/2,v)	$4(W)G_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,-w,v)	$4GP_{1}^{S}(1)$	Spin Splitting

4z

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)	E:(1/2,1/2,1/2)	$(R)E_{1}^{S,-}(2)⊕(R)E_{2}^{S,+}(2) $
S:(1/2,1/2,0)	A:(1/2,0,1/2)	$(S)A_{1}^{S,+}(2)⊕2(S)A_{2}^{S,-}(2) $
T:(0,1/2,-1/2)	D:(0,1/2,1/2)	$(T)D_{1}^{S,-}(2)⊕2(T)D_{2}^{S,+}(2) $
U:(1/2,0,-1/2)	C:(1/2,1/2,0)	$(U)C_{1}^{S,-}(2)⊕2(U)C_{2}^{S,+}(2) $
X:(1/2,0,0)	Y:(1/2,0,0)	$(X)Y_{1}^{S,+}(2)⊕2(X)Y_{2}^{S,-}(2) $
Y:(0,1/2,0)	B:(0,0,1/2)	$(Y)B_{1}^{S,+}(2)⊕2(Y)B_{2}^{S,-}(2) $
Z:(0,0,-1/2)	Z:(0,1/2,0)	$Z_{1}^{S,-}(2)⊕Z_{2}^{S,+}(2) $
A:(u,0,-1/2)	G:(u,1/2,0)	$2(A)G_{2}^{S}(2) $
B:(0,v,-1/2)	G:(0,1/2,v)	$2(B)G_{2}^{S}(2) $
C:(u,1/2,0)	F:(u,0,1/2)	$2(C)F_{1}^{S}(2) $
D:(1/2,v,0)	F:(1/2,0,v)	$2(D)F_{1}^{S}(2) $
Δ:(0,v,0)	F:(0,0,v)	$2(Δ)F_{1}^{S}(2) $
E:(u,1/2,-1/2)	G:(u,1/2,1/2)	$2(E)G_{2}^{S}(2) $
G:(1/2,0,w)	W:(1/2,-w,0)	$(G)W_{1}^{S}(2)⊕(G)W_{2}^{S}(2) $
H:(0,1/2,w)	V:(0,-w,1/2)	$(H)V_{1}^{S}(2)⊕(H)V_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
P:(1/2,v,-1/2)	G:(1/2,1/2,v)	$2(P)G_{2}^{S}(2) $
Q:(1/2,1/2,w)	U:(1/2,-w,1/2)	$(Q)U_{1}^{S}(2)⊕(Q)U_{2}^{S}(2) $
Σ:(u,0,0)	F:(u,0,0)	$2(Σ)F_{1}^{S}(2) $
K:(0,v,w)	GP:(0,-w,v)	$2(K)GP_{1}^{S}(2) $
L:(1/2,v,w)	GP:(1/2,-w,v)	$2(L)GP_{1}^{S}(2) $
M:(u,0,w)	GP:(u,-w,0)	$2(M)GP_{1}^{S}(2) $
N:(u,1/2,w)	GP:(u,-w,1/2)	$2(N)GP_{1}^{S}(2)$	Spin Splitting
V:(u,v,0)	F:(u,0,v)	$4(V)F_{1}^{S}(1)$	Spin Splitting
W:(u,v,-1/2)	G:(u,1/2,v)	$4(W)G_{2}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,-w,v)	$4GP_{1}^{S}(1)$	Spin Splitting

8A

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:(1/2,1/2,-1/2)	E:(1/2,1/2,1/2)	$(R)E_{1}^{S,+}(2)⊕(R)E_{1}^{S,-}(2)⊕(R)E_{2}^{S,+}(2)⊕(R)E_{2}^{S,-}(2) $
S:(1/2,1/2,0)	A:(1/2,0,1/2)	$(S)A_{1}^{S,+}(2)⊕(S)A_{1}^{S,-}(2)⊕(S)A_{2}^{S,+}(2)⊕(S)A_{2}^{S,-}(2) $
T:(0,1/2,-1/2)	D:(0,1/2,1/2)	$(T)D_{1}^{S,+}(2)⊕(T)D_{1}^{S,-}(2)⊕(T)D_{2}^{S,+}(2)⊕(T)D_{2}^{S,-}(2) $
U:(1/2,0,-1/2)	C:(1/2,1/2,0)	$(U)C_{1}^{S,+}(2)⊕(U)C_{1}^{S,-}(2)⊕(U)C_{2}^{S,+}(2)⊕(U)C_{2}^{S,-}(2) $
X:(1/2,0,0)	Y:(1/2,0,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{1}^{S,-}(2)⊕(X)Y_{2}^{S,+}(2)⊕(X)Y_{2}^{S,-}(2) $
Y:(0,1/2,0)	B:(0,0,1/2)	$(Y)B_{1}^{S,+}(2)⊕(Y)B_{1}^{S,-}(2)⊕(Y)B_{2}^{S,+}(2)⊕(Y)B_{2}^{S,-}(2) $
Z:(0,0,-1/2)	Z:(0,1/2,0)	$Z_{1}^{S,+}(2)⊕Z_{1}^{S,-}(2)⊕Z_{2}^{S,+}(2)⊕Z_{2}^{S,-}(2) $
A:(u,0,-1/2)	G:(u,1/2,0)	$2(A)G_{1}^{S}(2)⊕2(A)G_{2}^{S}(2) $
B:(0,v,-1/2)	G:(0,1/2,v)	$2(B)G_{1}^{S}(2)⊕2(B)G_{2}^{S}(2) $
C:(u,1/2,0)	F:(u,0,1/2)	$2(C)F_{1}^{S}(2)⊕2(C)F_{2}^{S}(2) $
D:(1/2,v,0)	F:(1/2,0,v)	$2(D)F_{1}^{S}(2)⊕2(D)F_{2}^{S}(2) $
Δ:(0,v,0)	F:(0,0,v)	$2(Δ)F_{1}^{S}(2)⊕2(Δ)F_{2}^{S}(2) $
E:(u,1/2,-1/2)	G:(u,1/2,1/2)	$2(E)G_{1}^{S}(2)⊕2(E)G_{2}^{S}(2) $
G:(1/2,0,w)	W:(1/2,-w,0)	$2(G)W_{1}^{S}(2)⊕2(G)W_{2}^{S}(2) $
H:(0,1/2,w)	V:(0,-w,1/2)	$2(H)V_{1}^{S}(2)⊕2(H)V_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2) $
P:(1/2,v,-1/2)	G:(1/2,1/2,v)	$2(P)G_{1}^{S}(2)⊕2(P)G_{2}^{S}(2) $
Q:(1/2,1/2,w)	U:(1/2,-w,1/2)	$2(Q)U_{1}^{S}(2)⊕2(Q)U_{2}^{S}(2) $
Σ:(u,0,0)	F:(u,0,0)	$2(Σ)F_{1}^{S}(2)⊕2(Σ)F_{2}^{S}(2) $
K:(0,v,w)	GP:(0,-w,v)	$4(K)GP_{1}^{S}(2) $
L:(1/2,v,w)	GP:(1/2,-w,v)	$4(L)GP_{1}^{S}(2) $
M:(u,0,w)	GP:(u,-w,0)	$4(M)GP_{1}^{S}(2) $
N:(u,1/2,w)	GP:(u,-w,1/2)	$4(N)GP_{1}^{S}(2) $
V:(u,v,0)	F:(u,0,v)	$4(V)F_{1}^{S}(1)⊕4(V)F_{2}^{S}(1)$	Spin Splitting
W:(u,v,-1/2)	G:(u,1/2,v)	$4(W)G_{1}^{S}(1)⊕4(W)G_{2}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,-w,v)	$8GP_{1}^{S}(1)$	Spin Splitting