# Difference between revisions of "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 $p$-group. See [Cr11] or [AKO11]. The fusion system given by a finite group $G$ on a Sylow $p$-subgroup $P$ is written $\mathcal{F}_P(G)$. The Fusion system for a block $B$ (sometimes calle the Brauer category) is defined with respect to a maximal subpair $(D,b_D)$, and is written $\mathcal{F}_{(D,b_D)}(G,B)$.

### Height of an irreducible character

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

### Index p covering blocks

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

### Isotypy

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

### # lifts / $\mathcal{O}$

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

### MNA(r,s)

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

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

### p'-index covered blocks

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

### p'-index covering blocks

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

### Picard group

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

### Possible Brauer tree (for a given cyclic defect group)

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

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

### Resistant p-group

A p-group $P$ is resistant if whenever $\mathcal{F}$ is a saturated fusion system on $P$, we have $\mathcal{F}=N_{\mathcal{F}}(P)$, or equivalently $\mathcal{F}=\mathcal{F}_P(G)$ for some finite group $G$ with $P$ 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 $B$ of a finite group (and more generally for a $G$-algebra) not only Morita equivalent to $B$ 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 $H \leq G$ such that $\forall g \in G \setminus N_G(H)$ we have $H^g\cap H=1$.