# Difference between revisions of "Notation"

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− | <math>(K,\mathcal{O},k)</math> is a <math>p</math>-modular system, where <math>\mathcal{O}</math> is a complete discrete valuation ring with algebraically closed residue field <math>k=\mathcal{O}/J(\mathcal{O})</math> and <math>K</math> is the field of fractions of <math>\mathcal{O}</math>, of characteristic zero. | + | <math>(K,\mathcal{O},k)</math> is a <math>p</math>-modular system, where <math>\mathcal{O}</math> is a complete discrete valuation ring with algebraically closed residue field <math>k=\mathcal{O}/J(\mathcal{O})</math> and <math>K</math> is the field of fractions of <math>\mathcal{O}</math>, of characteristic zero. When we need to make a consistent choice of <math>(K,\mathcal{O},k)</math> we take <math>k</math> to be the algebraic closure of the field with <math>p</math> elements and <math>\mathcal{O}</math> to be the ring of Witt vectors for <math>k</math>. This has the disadvantage that for <math>G</math> a finite group <math>KG</math> need not contain the primitive character idempotents, but this condition can usually be avoided. In general however the choice of <math>\mathcal{O}</math> is not consistent across the literature and some care has to be taken. |

In the below, <math>G</math> is a finite group and <math>B</math> is a block of <math>\mathcal{O}G</math>. If it is clear from context, <math>B</math> may also mean the corresponding block of <math>kG</math>. When it is not otherwise clear from context <math>kB</math> will refer to the block of <math>kG</math>. | In the below, <math>G</math> is a finite group and <math>B</math> is a block of <math>\mathcal{O}G</math>. If it is clear from context, <math>B</math> may also mean the corresponding block of <math>kG</math>. When it is not otherwise clear from context <math>kB</math> will refer to the block of <math>kG</math>. | ||

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− | |<math>k(B)</math> || Number of irreducible characters in <math>B</math> || | + | |<math>k(B)</math> || Number of irreducible characters in <math>B</math>, equal to <math>\dim_k(Z(kB))</math> || |

+ | |- | ||

+ | |<math>k_i(B)</math> || Number of irreducible characters in <math>B</math> of height <math>i</math> || | ||

|- | |- | ||

|<math>l(B)</math> || Number of isomorphism classes of simple <math>B</math>-modules || | |<math>l(B)</math> || Number of isomorphism classes of simple <math>B</math>-modules || |

## Latest revision as of 18:45, 9 December 2019

[math](K,\mathcal{O},k)[/math] is a [math]p[/math]-modular system, where [math]\mathcal{O}[/math] is a complete discrete valuation ring with algebraically closed residue field [math]k=\mathcal{O}/J(\mathcal{O})[/math] and [math]K[/math] is the field of fractions of [math]\mathcal{O}[/math], of characteristic zero. When we need to make a consistent choice of [math](K,\mathcal{O},k)[/math] we take [math]k[/math] to be the algebraic closure of the field with [math]p[/math] elements and [math]\mathcal{O}[/math] to be the ring of Witt vectors for [math]k[/math]. This has the disadvantage that for [math]G[/math] a finite group [math]KG[/math] need not contain the primitive character idempotents, but this condition can usually be avoided. In general however the choice of [math]\mathcal{O}[/math] is not consistent across the literature and some care has to be taken.

In the below, [math]G[/math] is a finite group and [math]B[/math] is a block of [math]\mathcal{O}G[/math]. If it is clear from context, [math]B[/math] may also mean the corresponding block of [math]kG[/math]. When it is not otherwise clear from context [math]kB[/math] will refer to the block of [math]kG[/math].

[math]k(B)[/math] | Number of irreducible characters in [math]B[/math], equal to [math]\dim_k(Z(kB))[/math] | |

[math]k_i(B)[/math] | Number of irreducible characters in [math]B[/math] of height [math]i[/math] | |

[math]l(B)[/math] | Number of isomorphism classes of simple [math]B[/math]-modules | |

[math]{\rm mf_k(B)}[/math] | The Morita-Frobenius number of [math]kB[/math] | [Ke04] |

[math]{\rm mf_\mathcal{O}(B)}[/math] | The [math]\mathcal{O}[/math]-Morita Frobenius number | |

[math]{\rm Pic}_\mathcal{O}(B)[/math] | The Picard group of [math]B[/math] | |

[math]{\rm Pic}_k(B)[/math] | The Picard group of [math]kB[/math] | |

[math]\mathcal{T}(B)[/math] | The subgroup of [math]{\rm Pic}_\mathcal{O}(B)[/math] consisting of trivial source bimodules | [BKL18] |

[math]\mathcal{L}(B)[/math] | The subgroup of [math]{\rm Pic}_\mathcal{O}(B)[/math] consisting of linear source bimodules | [BKL18] |

[math]\mathcal{E}(B)[/math] | The subgroup of [math]{\rm Pic}_\mathcal{O}(B)[/math] consisting of endopermutation source bimodules | [BKL18] |

[math]M(x,y,z)[/math] | A [math]k[/math]-Morita equivalence class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class. |