# M(27,1,3)

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M(27,1,3) - [math]B_0(kPSL_2(53))[/math]

Representative: | [math]B_0(kPSL_2(53))[/math] |
---|---|

Defect groups: | [math]C_{27}[/math] |

Inertial quotients: | [math]C_2[/math] |

[math]k(B)=[/math] | 15 |

[math]l(B)=[/math] | 2 |

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

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

Cartan matrix: | [math]\left( \begin{array}{cc} 2 & 1 \\ 1 & 14 \\ \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]B_0(\mathcal{O} PSL_2(53))[/math] |

Decomposition matrices: | [math]\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ \vdots & \vdots \\ 0 & 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? | Yes |

Source algebra reps: | |

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

[math]k[/math]-derived equivalent to: | M(27,1,2) |

[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}}} |

## Contents

## Basic algebra

**Quiver:** a:<1,2>, b:<2,1>, c:<2,2>

**Relations w.r.t. [math]k[/math]:** ac=cb=ba-c^{13}=0

## Other notatable representatives

## Covering blocks and covered blocks

## Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, S_2[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{cc} \begin{array}{c} S_1 \\ S_2 \\ S_1 \\ \end{array}, & \begin{array}{ccc} & S_2 & \\ S_1 & & \begin{array}{c} S_2 \\ S_2 \\ \vdots \\ S_2 \\ \end{array} \\ & S_2 & \\ \end{array} \end{array} [/math]

## Irreducible characters

All irreducible characters have height zero.