# Difference between revisions of "M(9,1,1)"

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## Revision as of 13:29, 9 September 2018

Representative: | [math]kC_9[/math] |
---|---|

Defect groups: | [math]C_9[/math] |

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

[math]k(B)=[/math] | 9 |

[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} 9 \\ \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} C_9[/math] |

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

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

[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | [math]\mathcal{L}(B)=C_9:C_6[/math] |

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

Source algebras known? | Yes |

Source algebra reps: | [math]kC_9[/math] |

[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: | {{{coveringblocks}}} |

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

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

These are nilpotent blocks.

## Contents

## Basic algebra

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

**Relations w.r.t. [math]k[/math]:** a^9=0

## Other notatable representatives

## Covering blocks and covered blocks

Let [math]N \triangleleft G[/math] with [math]p'[/math]-index and let [math]B[/math] be a block of [math]\mathcal{O} G[/math] covering a block [math]b[/math] of [math]\mathcal{O} N[/math].

If [math]b[/math] lies in M(9,1,1), then [math]B[/math] must lie in M(9,1,1) or M(9,1,2). For example consider the principal blocks of [math]C_9 \triangleleft D_{18}[/math].

If [math]B[/math] lies in M(9,1,1), then [math]b[/math] must lie in M(9,1,1) or M(9,1,2). Examples needed.

## 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 \\ \vdots \\ S_1 \\ \end{array} [/math]

## Irreducible characters

All irreducible characters have height zero.