Difference between revisions of "Glossary"

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(Trivial intersection)
(covering/covered block entries)
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Fusion system on a finite <math>p</math>-group. See [[References|[Cr11] or [AKO11]]]. The fusion system given by a finite group <math>G</math> on a Sylow <math>p</math>-subgroup <math>P</math> is written <math>\mathcal{F}_P(G)</math>.
 
Fusion system on a finite <math>p</math>-group. See [[References|[Cr11] or [AKO11]]]. The fusion system given by a finite group <math>G</math> on a Sylow <math>p</math>-subgroup <math>P</math> is written <math>\mathcal{F}_P(G)</math>.
  
=== # lifts / <math>\mathcal{O}</math> ===
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=== Index p covering blocks (table entry) ===
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 +
Fix a Morita equivalence class <math>M</math>. This lists Morita equivalence classes containing a block <math>B</math> of <math>kG</math> for some finite group <math>G</math> such that <math>B</math> covers a block <math>b</math> in <math>M</math> of <math>kN</math> for some normal subgroup <math>N</math> of <math>G</math> of index <math>p</math>.
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=== # lifts / <math>\mathcal{O}</math> (table entry) ===
  
 
The number of <math>\mathcal{O}</math>-Morita equivalence classes of blocks reducing to a representative of the given <math>k</math>-class.
 
The number of <math>\mathcal{O}</math>-Morita equivalence classes of blocks reducing to a representative of the given <math>k</math>-class.
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 +
=== p'-index covered blocks (table entry) ===
 +
 +
Fix a Morita equivalence class <math>M</math>. This lists Morita equivalence classes containing a block <math>b</math> of <math>kN</math> for some finite group <math>N</math> such that <math>b</math> is covered by a block <math>B</math> in <math>M</math> of <math>kG</math> for some finite group <math>G</math> containing <math>N</math> as a normal subgroup of prime index different to <math>p</math>.
 +
 +
=== p'-index covering blocks (table entry) ===
 +
 +
Fix a Morita equivalence class <math>M</math>. This lists Morita equivalence classes containing a block <math>B</math> of <math>kG</math> for some finite group <math>G</math> such that <math>B</math> covers a block <math>b</math> in <math>M</math> of <math>kN</math> for some normal subgroup <math>N</math> of <math>G</math> of prime index different to <math>p</math>.
  
 
=== Picard group ===
 
=== Picard group ===

Revision as of 12:21, 27 October 2018

This page will contain an alphabetical glossary of terms used.

Basic Morita/stable equivalence

Morita/stable equivalence of blocks induced by a bimodule which has endopermutation source.

CFSG

The classification of finite simple groups.

Fusion system

Fusion system on a finite [math]p[/math]-group. See [Cr11] or [AKO11]. The fusion system given by a finite group [math]G[/math] on a Sylow [math]p[/math]-subgroup [math]P[/math] is written [math]\mathcal{F}_P(G)[/math].

Index p covering blocks (table entry)

Fix a Morita equivalence class [math]M[/math]. This lists Morita equivalence classes containing a block [math]B[/math] of [math]kG[/math] for some finite group [math]G[/math] such that [math]B[/math] covers a block [math]b[/math] in [math]M[/math] of [math]kN[/math] for some normal subgroup [math]N[/math] of [math]G[/math] of index [math]p[/math].

# lifts / [math]\mathcal{O}[/math] (table entry)

The number of [math]\mathcal{O}[/math]-Morita equivalence classes of blocks reducing to a representative of the given [math]k[/math]-class.

p'-index covered blocks (table entry)

Fix a Morita equivalence class [math]M[/math]. This lists Morita equivalence classes containing a block [math]b[/math] of [math]kN[/math] for some finite group [math]N[/math] such that [math]b[/math] is covered by a block [math]B[/math] in [math]M[/math] of [math]kG[/math] for some finite group [math]G[/math] containing [math]N[/math] as a normal subgroup of prime index different to [math]p[/math].

p'-index covering blocks (table entry)

Fix a Morita equivalence class [math]M[/math]. This lists Morita equivalence classes containing a block [math]B[/math] of [math]kG[/math] for some finite group [math]G[/math] such that [math]B[/math] covers a block [math]b[/math] in [math]M[/math] of [math]kN[/math] for some normal subgroup [math]N[/math] of [math]G[/math] of prime index different to [math]p[/math].

Picard group

Let [math]R[/math] be a commutative ring and [math]A[/math] an [math]R[/math]-algebra. The Picard group [math]{\rm Pic}_R(A)[/math] has elements isomorphism classes of [math]A[/math]-[math]A[/math]-bimodules inducing a Morita equivalence, with multiplication given by taking tensor products over [math]A[/math].

Possible Brauer tree (for a given cyclic defect group)

Fix a cyclic group [math]P[/math] of order [math]p^n[/math]. A block with defect group [math]P[/math] has inertial index [math]e[/math] a divisor of [math]p-1[/math]. The number of irreducible characters in the block is [math]e+\frac{|P|-1}{e}[/math]. The exceptional vertex has multiplicity [math]\frac{|P|-1}{e}[/math].

The possibile Brauer trees (for [math]P[/math] and [math]e[/math] a divisor of [math]p-1[/math]) are the Brauer trees whose vertex multiplicities add to [math]e+\frac{|P|-1}{e}[/math] where the exceptional vertex multiplicity is [math]\frac{|P|-1}{e}[/math] and non-exceptional vertices are regarded as having multiplicity [math]1[/math].

Splendid equivalence

May apply to Morita equivalences, stable equivalences and derived equivalences. See 9.7 an 9.8 of [Li18d]. It means roughly equivalences given by (complexes of) trivial source bimodules.

Trivial intersection subgroup

A subgroup [math]H \leq G[/math] such that [math]\forall g \in G \setminus N_G(H)[/math] we have [math]H^g\cap H=1[/math].