# Difference between revisions of "Glossary"

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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]]]. | ||

=== Source algebra === | === Source algebra === |

## Latest revision as of 23:02, 18 November 2020

## Contents

- 1 Basic Morita/stable equivalence
- 2 CFSG
- 3 Fusion system
- 4 Height of an irreducible character
- 5 Index p covering blocks
- 6 Isotypy
- 7 # lifts / [math]\mathcal{O}[/math]
- 8 MNA(r,s)
- 9 p'-index covered blocks
- 10 p'-index covering blocks
- 11 Picard group
- 12 Possible Brauer tree (for a given cyclic defect group)
- 13 Resistant p-group
- 14 Source algebra
- 15 Splendid equivalence
- 16 Trivial intersection subgroup

### 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].