# M(8,3,1)

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M(8,3,1) - [math]kD_8[/math]

Representative: | [math]kD_8[/math] |
---|---|

Defect groups: | [math]D_8[/math] |

Inertial quotients: | [math]1[/math] |

[math]k(B)=[/math] | 5 |

[math]l(B)=[/math] | 1 |

[math]{\rm mf}_k(B)=[/math] | 1 |

[math]{\rm Pic}_k(B)=[/math] | |

Cartan matrix: | [math]\left( \begin{array}{c} 8 \\ \end{array} \right)[/math] |

Defect group Morita invariant? | Yes |

Inertial quotient Morita invariant? | Yes |

[math]\mathcal{O}[/math]-Morita classes known? | Yes |

[math]\mathcal{O}[/math]-Morita classes: | [math]\mathcal{O}D_8[/math] |

Decomposition matrices: | [math]\left( \begin{array}{c} 2 \\ 1 \\ 1 \\ 1 \\ 1 \\ \end{array}\right)[/math] |

[math]{\rm mf}_\mathcal{O}(B)=[/math] | 1 |

[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | |

[math]PI(B)=[/math] | {{{PIgroup}}} |

Source algebras known? | No |

Source algebra reps: | |

[math]k[/math]-derived equiv. classes known? | Yes |

[math]k[/math]-derived equivalent to: | Forms a derived equivalence class |

[math]\mathcal{O}[/math]-derived equiv. classes known? | Yes |

[math]p'[/math]-index covering blocks: | |

[math]p'[/math]-index covered blocks: | |

Index [math]p[/math] covering blocks: | {{{pcoveringblocks}}} |

These are nilpotent blocks.

## Contents

## Basic algebra

**Quiver:** a:<1,1>, b:<1,1>

**Relations w.r.t. [math]k[/math]:** a^2=b^2=(ab)^2+(ba)^2=0

## Other notatable representatives

## Projective indecomposable modules

Labelling the unique simple [math]B[/math]-module by [math]S_1[/math], the unique projective indecomposable module has Loewy structure as follows:

[math]\begin{array}{c} S_1 \\ S_1 S_1 \\ S_1 S_1 \\ S_1 S_1 \\ S_1 \\ \end{array} [/math]

## Irreducible characters

[math]k_0(B)=4, k_1(B)=1[/math]