nLab character table of 2T

linear representation theory of binary tetrahedral group $2 T$

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group order: $\vert 2T\vert = 24$

conjugacy classes:	1	-1	$i$	a	b	c	d
their cardinality:	1	1	6	4	4	4	4

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let $\zeta_3$ be a third root of unity, $(\zeta_3)^3 = 1$

e.g. $\zeta_3 = \tfrac{1}{2}(-1 + \sqrt{3} i)$, notice that $\zeta_3 + \zeta_3^2 = 1$

character table over the complex numbers $\mathbb{C}$

irrep	1	-1	$i$	a	b	c	d
$\rho_1$	1	1	1	1	1	1	1
$\rho_2$	1	1	1	$\zeta_3$	$\zeta_3^2$	$\zeta_3^2$	$\zeta_3$
$\rho_2^\ast$	1	1	1	$\zeta^2_3$	$\zeta_3$	$\zeta_3$	$\zeta_3^2$
$\rho_3$	3	3	-1	0	0	0	0
$\rho_4$	2	-2	0	$\zeta_3$	$\zeta_3^2$	$-\zeta_3^2$	$-\zeta_3$
$\rho_4^\ast$	2	-2	0	$\zeta_3^2$	$\zeta_3$	$-\zeta_3$	$-\zeta_3^2$
$\rho_5$	2	-2	0	1	1	-1	-1

character table over the real numbers $\mathbb{R}$

irrep	1	-1	$i$	a	b	c	d
$\rho_1$	1	1	1	1	1	1	1
$\rho_2 \oplus \rho_2^\ast$	2	2	2	-1	-1	-1	-1
$\rho_3$	3	3	-1	0	0	0	0
$\rho_4 \oplus \rho_4^\ast$	4	-4	0	-1	-1	1	1
$\rho_5 \oplus \rho_5$	4	-4	0	2	2	-2	-2

References

• GroupNames, SL(2,3)

• James Montaldi, Real representations – Binary cubic – 2T

• Bockland, Character tables and McKay quivers (pdf)

category: character tables