Information of spin space group $C_a{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$ ( nSSG:12.10.2.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
4a	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}/{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,0 | 0,0,z)
4b	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}/{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,1/4,0 | 0,0,0)
4c	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}/{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,1/2 | 0,0,z)
4d	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}/{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(1/4,0,0 | 0,0,0)
4e	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}/{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(1/4,1/4,0 | 0,0,0)
4f	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}/{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,1/4,1/2 | 0,0,0)
4g	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}/{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(1/4,0,1/2 | 0,0,0)
4h	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}/{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(1/4,1/4,1/2 | 0,0,0)
8i	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,b,0 | 0,0,z)
8j	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(1/4,b,0 | 0,0,0)
8k	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,b,1/2 | 0,0,z)
8l	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(1/4,b,1/2 | 0,0,0)
8m	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,0,c | 0,0,z)
8n	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,1/4,c | 0,0,0)
16o	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,b,c | 0,0,z)

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group
A:(0,0,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
Γ:(0,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
M:(0,1,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
Y:(0,1,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
L:(1/2,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
V:(1/2,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
B:(u,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
Λ:(0,v,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
U:(0,v,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
GP:(u,v,w)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{-1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$

Spin Brillouin Zone

2/mC

Band Representations (Magnon)

4a

k-vector	Ag↑G(2)
A:(0,0,1/2)	$A_1^{S,+}(2)$
Γ:(0,0,0)	${\Gamma }_1^{S,+}(2)$
M:(0,1,1/2)	$M_1^{S,+}(2)$
Y:(0,1,0)	$Y_1^{S,+}(2)$
L:(1/2,1/2,1/2)	$L_1^{S,+}(2)$
V:(1/2,1/2,0)	$V_1^{S,+}(2)$
B:(u,0,w)	$B_1^S(2)$
Λ:(0,v,0)	${\Lambda }_1^S(2)$
U:(0,v,1/2)	$U_1^S(2)$
GP:(u,v,w)	$G{P}_1^S(2)$

4c

k-vector	Ag↑G(2)
A:(0,0,1/2)	$A_2^{S,-}(2)$
Γ:(0,0,0)	${\Gamma }_1^{S,+}(2)$
M:(0,1,1/2)	$M_2^{S,-}(2)$
Y:(0,1,0)	$Y_1^{S,+}(2)$
L:(1/2,1/2,1/2)	$L_1^{S,-}(2)$
V:(1/2,1/2,0)	$V_1^{S,+}(2)$
B:(u,0,w)	$B_1^S(2)$
Λ:(0,v,0)	${\Lambda }_1^S(2)$
U:(0,v,1/2)	$U_2^S(2)$
GP:(u,v,w)	$G{P}_1^S(2)$

8i

k-vector	A↑G(4)
A:(0,0,1/2)	$A_1^{S,+}(2)\oplus A_1^{S,-}(2)$
Γ:(0,0,0)	${\Gamma }_1^{S,+}(2)\oplus {\Gamma }_1^{S,-}(2)$
M:(0,1,1/2)	$M_1^{S,+}(2)\oplus M_1^{S,-}(2)$
Y:(0,1,0)	$Y_1^{S,+}(2)\oplus Y_1^{S,-}(2)$
L:(1/2,1/2,1/2)	$L_1^{S,+}(2)\oplus L_1^{S,-}(2)$
V:(1/2,1/2,0)	$V_1^{S,+}(2)\oplus V_1^{S,-}(2)$
B:(u,0,w)	$B_1^S(2)\oplus B_2^S(2)$
Λ:(0,v,0)	$2{\Lambda }_1^S(2)$
U:(0,v,1/2)	$2U_1^S(2)$
GP:(u,v,w)	$2G{P}_1^S(2)$

8k

k-vector	A↑G(4)
A:(0,0,1/2)	$A_2^{S,+}(2)\oplus A_2^{S,-}(2)$
Γ:(0,0,0)	${\Gamma }_1^{S,+}(2)\oplus {\Gamma }_1^{S,-}(2)$
M:(0,1,1/2)	$M_2^{S,+}(2)\oplus M_2^{S,-}(2)$
Y:(0,1,0)	$Y_1^{S,+}(2)\oplus Y_1^{S,-}(2)$
L:(1/2,1/2,1/2)	$L_1^{S,+}(2)\oplus L_1^{S,-}(2)$
V:(1/2,1/2,0)	$V_1^{S,+}(2)\oplus V_1^{S,-}(2)$
B:(u,0,w)	$B_1^S(2)\oplus B_2^S(2)$
Λ:(0,v,0)	$2{\Lambda }_1^S(2)$
U:(0,v,1/2)	$2U_2^S(2)$
GP:(u,v,w)	$2G{P}_1^S(2)$

8m

k-vector	A'↑G(4)
A:(0,0,1/2)	$A_1^{S,+}(2)\oplus A_2^{S,-}(2)$
Γ:(0,0,0)	${\Gamma }_1^{S,+}(2)\oplus {\Gamma }_2^{S,-}(2)$
M:(0,1,1/2)	$M_1^{S,+}(2)\oplus M_2^{S,-}(2)$
Y:(0,1,0)	$Y_1^{S,+}(2)\oplus Y_2^{S,-}(2)$
L:(1/2,1/2,1/2)	$L_1^{S,+}(2)\oplus L_1^{S,-}(2)$
V:(1/2,1/2,0)	$V_1^{S,+}(2)\oplus V_1^{S,-}(2)$
B:(u,0,w)	$2B_1^S(2)$
Λ:(0,v,0)	${\Lambda }_1^S(2)\oplus {\Lambda }_2^S(2)$
U:(0,v,1/2)	$U_1^S(2)\oplus U_2^S(2)$
GP:(u,v,w)	$2G{P}_1^S(2)$

16o

k-vector	A↑G(8)
A:(0,0,1/2)	$A_1^{S,+}(2)\oplus A_1^{S,-}(2)\oplus A_2^{S,+}(2)\oplus A_2^{S,-}(2)$
Γ:(0,0,0)	${\Gamma }_1^{S,+}(2)\oplus {\Gamma }_1^{S,-}(2)\oplus {\Gamma }_2^{S,+}(2)\oplus {\Gamma }_2^{S,-}(2)$
M:(0,1,1/2)	$M_1^{S,+}(2)\oplus M_1^{S,-}(2)\oplus M_2^{S,+}(2)\oplus M_2^{S,-}(2)$
Y:(0,1,0)	$Y_1^{S,+}(2)\oplus Y_1^{S,-}(2)\oplus Y_2^{S,+}(2)\oplus Y_2^{S,-}(2)$
L:(1/2,1/2,1/2)	$2L_1^{S,+}(2)\oplus 2L_1^{S,-}(2)$
V:(1/2,1/2,0)	$2V_1^{S,+}(2)\oplus 2V_1^{S,-}(2)$
B:(u,0,w)	$2B_1^S(2)\oplus 2B_2^S(2)$
Λ:(0,v,0)	$2{\Lambda }_1^S(2)\oplus 2{\Lambda }_2^S(2)$
U:(0,v,1/2)	$2U_1^S(2)\oplus 2U_2^S(2)$
GP:(u,v,w)	$4G{P}_1^S(2)$