Difference between revisions of "Glossary"
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May apply to Morita equivalences, stable equivalences and derived equivalences. See 9.7 an 9.8 of [[References|[Li18d]]]. It means roughly equivalences given by (complexes of) trivial source bimodules. | May apply to Morita equivalences, stable equivalences and derived equivalences. See 9.7 an 9.8 of [[References|[Li18d]]]. It means roughly equivalences given by (complexes of) trivial source bimodules. | ||
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| + | === Trivial intersection subgroup === | ||
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| + | 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>. | ||
Revision as of 09:08, 14 October 2018
This page will contain an alphabetical glossary of terms used.
Contents
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].
# 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.
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].
