character table of 2T

**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**

category: character tables

Last revised on October 8, 2018 at 06:22:24. See the history of this page for a list of all contributions to it.