GL(2,3)

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

The finite general linear group $GL(2,\mathbb{3})$ for coefficients the prime field $\mathbb{F}_3$.

The character table for $GL(2,\mathbb{F}_3)$ over the complex numbers looks like that of the binary octahedral group. The only difference is the Schur indices. Hence the character tables over the real numbers do differ.

**linear representation theory of the finite general linear group GL(2,3)**

$\,$

group order: ${\vert GL(2,3)\vert} = 48$

conjugacy classes: | 1 | 2A | 2B | 3 | 4 | 6 | 8A | 8B |
---|---|---|---|---|---|---|---|---|

their cardinality: | 1 | 1 | 12 | 8 | 6 | 8 | 6 | 6 |

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

irrep | 1 | 2A | 2B | 3 | 4 | 6 | 8A | 8B |
---|---|---|---|---|---|---|---|---|

$\rho_1$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

$\rho_2$ | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 |

$\rho_3$ | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 0 |

$\rho_4$ | 2 | -2 | 0 | -1 | 0 | 1 | $-\sqrt{-2}$ | $\sqrt{-2}$ |

$\rho_5$ | 2 | -2 | 0 | -1 | 0 | 1 | $\sqrt{-2}$ | $-\sqrt{-2}$ |

$\rho_6$ | 3 | 3 | -1 | 0 | -1 | 0 | 1 | 1 |

$\rho_7$ | 3 | 3 | 1 | 0 | -1 | 0 | -1 | -1 |

$\rho_8$ | 4 | -4 | 0 | 1 | 0 | -1 | 0 | 0 |

**character table over the rational numbers $\mathbb{Q}$ and real numbers $\mathbb{R}$**

irrep | 1 | 2A | 2B | 3 | 4 | 6 | 8A | 8B |
---|---|---|---|---|---|---|---|---|

$\rho_1$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

$\rho_2$ | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 |

$\rho_3$ | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 0 |

$\rho_4 \oplus \rho_5$ | 4 | -4 | 0 | -2 | 0 | 2 | 0 | 0 |

$\rho_6$ | 3 | 3 | -1 | 0 | -1 | 0 | 1 | 1 |

$\rho_7$ | 3 | 3 | 1 | 0 | -1 | 0 | -1 | -1 |

$\rho_8$ | 4 | -4 | 0 | 1 | 0 | -1 | 0 | 0 |

**References**

category: character tables