Glossary

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].

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].