Difference between revisions of "Glossary"

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=== Fusion system ===
 
=== Fusion system ===
  
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>.
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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>. The Fusion system for a block <math>B</math> (sometimes calle the Brauer category) is defined with respect to a maximal subpair <math>(D,b_D)</math>, and is written <math>\mathcal{F}_{(D,b_D)}(G,B)</math>.
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=== Height of an irreducible character ===
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An irreducible character <math>\chi</math> in a block <math>B</math> of <math>\mathcal{O} G</math> with defect group <math>D</math> has height <math>h</math> if <math>\chi(1)_p=p^h[G:D]_p</math>.  
  
 
=== Index p covering blocks ===
 
=== Index p covering blocks ===
  
 
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>.
 
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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=== Isotypy ===
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A family of perfect isometries resulting from the existence of a splendid derived equivalence. Introduced by Broué. See [[References#L|[Li18d,9.5]]].
  
 
=== # lifts / <math>\mathcal{O}</math> ===
 
=== # lifts / <math>\mathcal{O}</math> ===
  
 
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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=== MNA(r,s) ===
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A class of minimal nonabelian <math>2</math>-groups, that is nonabelian <math>2</math>-groups such that every proper subgroup is abelian. For <math>r \geq s \geq 1</math>
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\[ MNA(r,s) = \langle x,y|x^{2^r}=y^{2^s}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle . \]
  
 
=== p'-index covered blocks ===
 
=== p'-index covered blocks ===
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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>.
 
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>.
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=== Resistant p-group ===
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A p-group <math>P</math> is resistant if whenever <math>\mathcal{F}</math> is a saturated fusion system on <math>P</math>, we have <math>\mathcal{F}=N_{\mathcal{F}}(P)</math>, or equivalently <math>\mathcal{F}=\mathcal{F}_P(G)</math> for some finite group <math>G</math> with <math>P</math> as a normal Sylow p-subgroup. Resistant p-groups were introduced in [[References#S|[St02]]] in terms of fusion systems for groups, and for arbitrary saturated fusion systems in [[References#S|[St06]]].
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=== Source algebra ===
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Introduced by Puig, these are subalgebras of a block <math>B</math> of a finite group (and more generally for a <math>G</math>-algebra) not only Morita equivalent to <math>B</math> but also determining the fusion system. See [[References#L|[Li18c,5.6.12]]].
  
 
=== Splendid equivalence ===
 
=== Splendid equivalence ===

Latest revision as of 22:02, 18 November 2020

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]. The Fusion system for a block [math]B[/math] (sometimes calle the Brauer category) is defined with respect to a maximal subpair [math](D,b_D)[/math], and is written [math]\mathcal{F}_{(D,b_D)}(G,B)[/math].

Height of an irreducible character

An irreducible character [math]\chi[/math] in a block [math]B[/math] of [math]\mathcal{O} G[/math] with defect group [math]D[/math] has height [math]h[/math] if [math]\chi(1)_p=p^h[G:D]_p[/math].

Index p covering blocks

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].

Isotypy

A family of perfect isometries resulting from the existence of a splendid derived equivalence. Introduced by Broué. See [Li18d,9.5].

# lifts / [math]\mathcal{O}[/math]

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

MNA(r,s)

A class of minimal nonabelian [math]2[/math]-groups, that is nonabelian [math]2[/math]-groups such that every proper subgroup is abelian. For [math]r \geq s \geq 1[/math]

\[ MNA(r,s) = \langle x,y|x^{2^r}=y^{2^s}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle . \]

p'-index covered blocks

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

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].

Resistant p-group

A p-group [math]P[/math] is resistant if whenever [math]\mathcal{F}[/math] is a saturated fusion system on [math]P[/math], we have [math]\mathcal{F}=N_{\mathcal{F}}(P)[/math], or equivalently [math]\mathcal{F}=\mathcal{F}_P(G)[/math] for some finite group [math]G[/math] with [math]P[/math] as a normal Sylow p-subgroup. Resistant p-groups were introduced in [St02] in terms of fusion systems for groups, and for arbitrary saturated fusion systems in [St06].

Source algebra

Introduced by Puig, these are subalgebras of a block [math]B[/math] of a finite group (and more generally for a [math]G[/math]-algebra) not only Morita equivalent to [math]B[/math] but also determining the fusion system. See [Li18c,5.6.12].

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].