linear representation theory of binary tetrahedral group $2 T$
$\,$
group order: $\vert 2T\vert = 24$
conjugacy classes: | 1 | -1 | $i$ | a | b | c | d |
---|---|---|---|---|---|---|---|
their cardinality: | 1 | 1 | 6 | 4 | 4 | 4 | 4 |
$\,$
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
Bockland, Character tables and McKay quivers (pdf)
Last revised on September 2, 2021 at 08:43:09. See the history of this page for a list of all contributions to it.