Information of spin space group $C\ce{^{\text{-}1}{2}}\ce{^{\text{-}1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:3.21.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
2a	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
2b	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,1/2,0 | 0,0,0)
2c	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/2,0,1/2 | 0,0,0)
2d	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 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,0,1/2 | 0,0,0)
4g	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(0,b,0 | 0,0,0)
4h	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(0,b,1/2 | 0,0,0)
4i	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,c | 0,0,z)
4j	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,z)
4k	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,c | 0,0,z)
8l	$\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}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
T:(1,0,1/2)	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Y:(1,0,0)	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Z:(0,0,1/2)	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
R:(1/2,1/2,1/2)	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $
S:(1/2,1/2,0)	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $
A:(u,0,1/2)	$\ce{^{2}{2}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
B:(0,v,1/2)	$\ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
H:(1,0,w)	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{2}{2}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
D:(1/2,1/2,w)	$\ce{^{1}{2}}\ce{^{\infty}{1}} $
K:(0,v,w)	$\ce{^{-1}{2}}\ce{^{\infty}{1}} $
M:(u,0,w)	$\ce{^{-1}{2}}\ce{^{\infty}{1}} $
P:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
Q:(u,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4i

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
R:(1/2,1/2,-1/2)	C:(1/2,1/2,0)	$2 (R)C_{1}^{S}(1)$	Spin Splitting
S:(1/2,1/2,0)	Y:(1/2,0,0)	$2 (S)Y_{1}^{S}(1)$	Spin Splitting
T:(0,1,-1/2)	E:(1/2,1/2,1/2)	$(T)E_{1}^{S}(2) $
Y:(0,1,0)	A:(1/2,0,1/2)	$(Y)A_{1}^{S}(2) $
Z:(0,0,-1/2)	Z:(0,1/2,0)	$Z_{1}^{S}(2) $
A:(u,0,-1/2)	G:(u/2,1/2,-u/2)	$(A)G_{1}^{S}(2) $
B:(0,v,-1/2)	G:(v/2,1/2,v/2)	$(B)G_{1}^{S}(2) $
D:(1/2,1/2,w)	W:(1/2,-w,0)	$2(D)W_{1}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	F:(v/2,0,v/2)	$(Δ)F_{1}^{S}(2) $
H:(0,1,w)	U:(1/2,-w,1/2)	$(H)U_{1}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2) $
Σ:(u,0,0)	F:(u/2,0,-u/2)	$(Σ)F_{1}^{S}(2) $
K:(0,v,w)	GP:(v/2,-w,v/2)	$(K)GP_{1}^{S}(2) $
M:(u,0,w)	GP:(u/2,-w,-u/2)	$(M)GP_{1}^{S}(2) $
P:(u,v,0)	F:(u/2+v/2,0,-u/2+v/2)	$2(P)F_{1}^{S}(1)$	Spin Splitting
Q:(u,v,-1/2)	G:(u/2+v/2,1/2,-u/2+v/2)	$2(Q)G_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u/2+v/2,-w,-u/2+v/2)	$2GP_{1}^{S}(1)$	Spin Splitting

4k

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{2}^{S}(2) $
R:(1/2,1/2,-1/2)	C:(1/2,1/2,0)	$2 (R)C_{2}^{S}(1)$	Spin Splitting
S:(1/2,1/2,0)	Y:(1/2,0,0)	$2 (S)Y_{2}^{S}(1)$	Spin Splitting
T:(0,1,-1/2)	E:(1/2,1/2,1/2)	$(T)E_{2}^{S}(2) $
Y:(0,1,0)	A:(1/2,0,1/2)	$(Y)A_{2}^{S}(2) $
Z:(0,0,-1/2)	Z:(0,1/2,0)	$Z_{1}^{S}(2) $
A:(u,0,-1/2)	G:(u/2,1/2,-u/2)	$(A)G_{1}^{S}(2) $
B:(0,v,-1/2)	G:(v/2,1/2,v/2)	$(B)G_{1}^{S}(2) $
D:(1/2,1/2,w)	W:(1/2,-w,0)	$2 (D)W_{2}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	F:(v/2,0,v/2)	$(Δ)F_{1}^{S}(2) $
H:(0,1,w)	U:(1/2,-w,1/2)	$(H)U_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2) $
Σ:(u,0,0)	F:(u/2,0,-u/2)	$(Σ)F_{1}^{S}(2) $
K:(0,v,w)	GP:(v/2,-w,v/2)	$(K)GP_{1}^{S}(2) $
M:(u,0,w)	GP:(u/2,-w,-u/2)	$(M)GP_{1}^{S}(2) $
P:(u,v,0)	F:(u/2+v/2,0,-u/2+v/2)	$2(P)F_{1}^{S}(1)$	Spin Splitting
Q:(u,v,-1/2)	G:(u/2+v/2,1/2,-u/2+v/2)	$2(Q)G_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u/2+v/2,-w,-u/2+v/2)	$2GP_{1}^{S}(1)$	Spin Splitting

4j

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
R:(1/2,1/2,-1/2)	C:(1/2,1/2,0)	$2 (R)C_{2}^{S}(1)$	Spin Splitting
S:(1/2,1/2,0)	Y:(1/2,0,0)	$2 (S)Y_{2}^{S}(1)$	Spin Splitting
T:(0,1,-1/2)	E:(1/2,1/2,1/2)	$(T)E_{1}^{S}(2) $
Y:(0,1,0)	A:(1/2,0,1/2)	$(Y)A_{1}^{S}(2) $
Z:(0,0,-1/2)	Z:(0,1/2,0)	$Z_{1}^{S}(2) $
A:(u,0,-1/2)	G:(u/2,1/2,-u/2)	$(A)G_{1}^{S}(2) $
B:(0,v,-1/2)	G:(v/2,1/2,v/2)	$(B)G_{1}^{S}(2) $
D:(1/2,1/2,w)	W:(1/2,-w,0)	$2(D)W_{2}^{S}(1)$	Spin Splitting
Δ:(0,v,0)	F:(v/2,0,v/2)	$(Δ)F_{1}^{S}(2) $
H:(0,1,w)	U:(1/2,-w,1/2)	$(H)U_{1}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2) $
Σ:(u,0,0)	F:(u/2,0,-u/2)	$(Σ)F_{1}^{S}(2) $
K:(0,v,w)	GP:(v/2,-w,v/2)	$(K)GP_{1}^{S}(2) $
M:(u,0,w)	GP:(u/2,-w,-u/2)	$(M)GP_{1}^{S}(2) $
P:(u,v,0)	F:(u/2+v/2,0,-u/2+v/2)	$2(P)F_{1}^{S}(1)$	Spin Splitting
Q:(u,v,-1/2)	G:(u/2+v/2,1/2,-u/2+v/2)	$2(Q)G_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u/2+v/2,-w,-u/2+v/2)	$2GP_{1}^{S}(1)$	Spin Splitting

8l

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