Information of spin space group $R{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-}\text{3}{\phantom{A}}_{\phantom{}}^{\phantom{\text{-}1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\text{-}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$ ( nSSG:148.166.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-b, a-b, c | x, y, z	{ 1 ‖ 3+001 | 0 }
-a+b, -a, c | x, y, z	{ 1 ‖ 3-001 | 0 }
-a, -b, -c | x, y, z	{ 1 ‖ -1 | 0 }
b, -a+b, -c | x, y, z	{ 1 ‖ -3+001 | 0 }
a-b, a, -c | x, y, z	{ 1 ‖ -3-001 | 0 }
a-b, -b, -c | -x, -y, -z	{ -1 ‖ 2100 | 0 }
b, a, -c | -x, -y, -z	{ -1 ‖ 2110 | 0 }
-a, -a+b, -c | -x, -y, -z	{ -1 ‖ 2010 | 0 }
-a+b, b, c | -x, -y, -z	{ -1 ‖ m100 | 0 }
-b, -a, c | -x, -y, -z	{ -1 ‖ m110 | 0 }
a, a-b, c | -x, -y, -z	{ -1 ‖ m010 | 0 }
a+2/3, b+1/3, c+1/3 | x, y, z	{ 1 ‖ 1 | 2/3 1/3 1/3 }
-b+2/3, a-b+1/3, c+1/3 | x, y, z	{ 1 ‖ 3+001 | 2/3 1/3 1/3 }
-a+b+2/3, -a+1/3, c+1/3 | x, y, z	{ 1 ‖ 3-001 | 2/3 1/3 1/3 }
-a+2/3, -b+1/3, -c+1/3 | x, y, z	{ 1 ‖ -1 | 2/3 1/3 1/3 }
b+2/3, -a+b+1/3, -c+1/3 | x, y, z	{ 1 ‖ -3+001 | 2/3 1/3 1/3 }
a-b+2/3, a+1/3, -c+1/3 | x, y, z	{ 1 ‖ -3-001 | 2/3 1/3 1/3 }
a-b+2/3, -b+1/3, -c+1/3 | -x, -y, -z	{ -1 ‖ 2100 | 2/3 1/3 1/3 }
b+2/3, a+1/3, -c+1/3 | -x, -y, -z	{ -1 ‖ 2110 | 2/3 1/3 1/3 }
-a+2/3, -a+b+1/3, -c+1/3 | -x, -y, -z	{ -1 ‖ 2010 | 2/3 1/3 1/3 }
-a+b+2/3, b+1/3, c+1/3 | -x, -y, -z	{ -1 ‖ m100 | 2/3 1/3 1/3 }
-b+2/3, -a+1/3, c+1/3 | -x, -y, -z	{ -1 ‖ m110 | 2/3 1/3 1/3 }
a+2/3, a-b+1/3, c+1/3 | -x, -y, -z	{ -1 ‖ m010 | 2/3 1/3 1/3 }
a+1/3, b+2/3, c+2/3 | x, y, z	{ 1 ‖ 1 | 1/3 2/3 2/3 }
-b+1/3, a-b+2/3, c+2/3 | x, y, z	{ 1 ‖ 3+001 | 1/3 2/3 2/3 }
-a+b+1/3, -a+2/3, c+2/3 | x, y, z	{ 1 ‖ 3-001 | 1/3 2/3 2/3 }
-a+1/3, -b+2/3, -c+2/3 | x, y, z	{ 1 ‖ -1 | 1/3 2/3 2/3 }
b+1/3, -a+b+2/3, -c+2/3 | x, y, z	{ 1 ‖ -3+001 | 1/3 2/3 2/3 }
a-b+1/3, a+2/3, -c+2/3 | x, y, z	{ 1 ‖ -3-001 | 1/3 2/3 2/3 }
a-b+1/3, -b+2/3, -c+2/3 | -x, -y, -z	{ -1 ‖ 2100 | 1/3 2/3 2/3 }
b+1/3, a+2/3, -c+2/3 | -x, -y, -z	{ -1 ‖ 2110 | 1/3 2/3 2/3 }
-a+1/3, -a+b+2/3, -c+2/3 | -x, -y, -z	{ -1 ‖ 2010 | 1/3 2/3 2/3 }
-a+b+1/3, b+2/3, c+2/3 | -x, -y, -z	{ -1 ‖ m100 | 1/3 2/3 2/3 }
-b+1/3, -a+2/3, c+2/3 | -x, -y, -z	{ -1 ‖ m110 | 1/3 2/3 2/3 }
a+1/3, a-b+2/3, c+2/3 | -x, -y, -z	{ -1 ‖ m010 | 1/3 2/3 2/3 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-b, a-b, c | -x, y, z	{ m ‖ 3+001 | 0 }
-a+b, -a, c | -x, y, z	{ m ‖ 3-001 | 0 }
-a, -b, -c | -x, y, z	{ m ‖ -1 | 0 }
b, -a+b, -c | -x, y, z	{ m ‖ -3+001 | 0 }
a-b, a, -c | -x, y, z	{ m ‖ -3-001 | 0 }
a-b, -b, -c | x, -y, -z	{ 2 ‖ 2100 | 0 }
b, a, -c | x, -y, -z	{ 2 ‖ 2110 | 0 }
-a, -a+b, -c | x, -y, -z	{ 2 ‖ 2010 | 0 }
-a+b, b, c | x, -y, -z	{ 2 ‖ m100 | 0 }
-b, -a, c | x, -y, -z	{ 2 ‖ m110 | 0 }
a, a-b, c | x, -y, -z	{ 2 ‖ m010 | 0 }
a+2/3, b+1/3, c+1/3 | -x, y, z	{ m ‖ 1 | 2/3 1/3 1/3 }
-b+2/3, a-b+1/3, c+1/3 | -x, y, z	{ m ‖ 3+001 | 2/3 1/3 1/3 }
-a+b+2/3, -a+1/3, c+1/3 | -x, y, z	{ m ‖ 3-001 | 2/3 1/3 1/3 }
-a+2/3, -b+1/3, -c+1/3 | -x, y, z	{ m ‖ -1 | 2/3 1/3 1/3 }
b+2/3, -a+b+1/3, -c+1/3 | -x, y, z	{ m ‖ -3+001 | 2/3 1/3 1/3 }
a-b+2/3, a+1/3, -c+1/3 | -x, y, z	{ m ‖ -3-001 | 2/3 1/3 1/3 }
a-b+2/3, -b+1/3, -c+1/3 | x, -y, -z	{ 2 ‖ 2100 | 2/3 1/3 1/3 }
b+2/3, a+1/3, -c+1/3 | x, -y, -z	{ 2 ‖ 2110 | 2/3 1/3 1/3 }
-a+2/3, -a+b+1/3, -c+1/3 | x, -y, -z	{ 2 ‖ 2010 | 2/3 1/3 1/3 }
-a+b+2/3, b+1/3, c+1/3 | x, -y, -z	{ 2 ‖ m100 | 2/3 1/3 1/3 }
-b+2/3, -a+1/3, c+1/3 | x, -y, -z	{ 2 ‖ m110 | 2/3 1/3 1/3 }
a+2/3, a-b+1/3, c+1/3 | x, -y, -z	{ 2 ‖ m010 | 2/3 1/3 1/3 }
a+1/3, b+2/3, c+2/3 | -x, y, z	{ m ‖ 1 | 1/3 2/3 2/3 }
-b+1/3, a-b+2/3, c+2/3 | -x, y, z	{ m ‖ 3+001 | 1/3 2/3 2/3 }
-a+b+1/3, -a+2/3, c+2/3 | -x, y, z	{ m ‖ 3-001 | 1/3 2/3 2/3 }
-a+1/3, -b+2/3, -c+2/3 | -x, y, z	{ m ‖ -1 | 1/3 2/3 2/3 }
b+1/3, -a+b+2/3, -c+2/3 | -x, y, z	{ m ‖ -3+001 | 1/3 2/3 2/3 }
a-b+1/3, a+2/3, -c+2/3 | -x, y, z	{ m ‖ -3-001 | 1/3 2/3 2/3 }
a-b+1/3, -b+2/3, -c+2/3 | x, -y, -z	{ 2 ‖ 2100 | 1/3 2/3 2/3 }
b+1/3, a+2/3, -c+2/3 | x, -y, -z	{ 2 ‖ 2110 | 1/3 2/3 2/3 }
-a+1/3, -a+b+2/3, -c+2/3 | x, -y, -z	{ 2 ‖ 2010 | 1/3 2/3 2/3 }
-a+b+1/3, b+2/3, c+2/3 | x, -y, -z	{ 2 ‖ m100 | 1/3 2/3 2/3 }
-b+1/3, -a+2/3, c+2/3 | x, -y, -z	{ 2 ‖ m110 | 1/3 2/3 2/3 }
a+1/3, a-b+2/3, c+2/3 | x, -y, -z	{ 2 ‖ m010 | 1/3 2/3 2/3 }

Wyckoff Positions

WP	Site symmetry	Representative
3a	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-3}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,0 | 0,0,0)
3b	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-3}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,1/2 | 0,0,0)
6c	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{3}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,z | 0,0,0)
9d	$.{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}/{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(1/2,0,1/2 | 0,0,0)
9e	$.{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}/{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(1/2,0,0 | 0,0,0)
18f	$.{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(x,0,0 | 0,0,0)
18g	$.{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(x,0,1/2 | 0,0,0)
18h	$.{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(x,-x,z | 0,0,0)
36i	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(x,y,z | 0,0,z)

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group
Γ:(0,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-3}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
T:(0,0,3/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-3}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
F:(0,1/2,1)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
L:(-1/2,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
Λ:(0,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{-3}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Σ:(u,-2*u,0)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Y:(u,u,3/2)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
C:(u,-u,w)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
GP:(u,v,w)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{-1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$

Spin Brillouin Zone

Band Representations (Magnon)

36i

k-vector	k-vector-G↑	A↑G(12)
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^{S,+}(2)\oplus {\Gamma }_1^{S,-}(2)\oplus {\Gamma }_2^{S,+}{\Gamma }_3^{S,+}(4)\oplus {\Gamma }_2^{S,-}{\Gamma }_3^{S,-}(4)$
F:(0,1/2,1)	F:(0,1/2,1)	$3F_1^{S,+}(2)\oplus 3F_1^{S,-}(2)$
L:(-1/2,1/2,1/2)	L:(-1/2,1/2,1/2)	$3L_1^{S,+}(2)\oplus 3L_1^{S,-}(2)$
T:(0,0,3/2)	T:(0,0,3/2)	$T_1^{S,+}(2)\oplus T_1^{S,-}(2)\oplus T_2^{S,+}{T}_3^{S,+}(4)\oplus T_2^{S,-}{T}_3^{S,-}(4)$
Λ:(0,0,w)	Λ:(0,0,w)	$2{\Lambda }_1^S(2)\oplus 2{\Lambda }_2^S{\Lambda }_3^S(4)$
Σ:(u,-2*u,0)	Σ:(u,-2*u,0)	$6{\Sigma }_1^S(2)$
Y:(u,u,3/2)	Y:(u,u,3/2)	$6Y_1^S(2)$
C:(u,-u,w)	C:(u,-u,w)	$6C_1^S(2)$
GP:(u,v,w)	GP:(u,v,w)	$12G{P}_1^S(1)$	Spin Splitting