# Statements of conjectures

## Donovan's conjecture

Donovan's conjecture can be stated over a field or a complete discrete valuation ring. It is usual to assume the field is algebraically closed, and it is no loss to assume it is the algebraic closure $k$ of the field with $p$ elements, as in the notation section.

### $k$-Donovan conjecture

Let $P$ be a finite $p$-group. Then there are only finitely many possible Morita equivalence classes for blocks of $kG$ for finite groups G with defect group isomorphic to $P$.

The situation with choice of discrete valuation ring is a little more complicated, as discussed in the notation section, and we take $\mathcal{O}$ to be the ring of Witt vectors for $k$.

### $\mathcal{O}$-Donovan conjecture

Let $P$ be a finite $p$-group. Then there are only finitely many possible Morita equivalence classes for blocks of $\mathcal{O} G$ for finite groups G with defect group isomorphic to $P$.

## Weak Donovan conjecture

This arose as a question of Brauer (where he asked whether the Cartan invariants (the entries of the Cartan matrix) are bounded by the order of a defect group). The answer to Brauer's precise question is no, but Donovan asked the more general question:

### Weak Donovan conjecture

Let $P$ be a finite $p$-group. Then there is $c=c(P) \in \mathbb{N}$ such that for all blocks of $kG$ for finite groups G with defect group isomorphic to $P$, the Cartan invariants are at most $c$.

By [Du04, pp. 19] the weak Donovan conjecture is equivalent to the following two conjectures together.

### Loewy length conjecture

Let $P$ be a finite $p$-group. Then there is $l=l(P) \in \mathbb{N}$ such that for all blocks of $kG$ for finite groups G with defect group isomorphic to $P$, the Loewy length is at most $l$.

### Ext space conjecture

Let $P$ be a finite $p$-group. Then there is $e=e(P) \in \mathbb{N}$ such that for all blocks $B$ of $kG$ for finite groups G with defect group isomorphic to $P$ and for all pairs $V, W$ of simple $B$-modules, $\dim_k({\rm Ext}_{kG}^1(V,W)) \leq e$

## Morita-Frobenius number conjectures

In [Ke05] Kessar showed that the $k$-Donovan conjecture is equivalent to the Weak Donovan conjecture together with the following.

### Kessar's conjecture

Let $P$ be a finite $p$-group-group. Then there is $m=m(P) \in \mathbb{N}$ such that if $G$ is a finite group and $B$ is a block of $kG$ with defect groups isomorphic to $P$, then ${\rm mf}_k(B) \leq m$.

This may be reformulated over $\mathcal{O}$. In [EEL18] it is shown that the $\mathcal{O}$-Donovan conjecture is equivalent to the Weak Donovan conjecture together with the following.

### $\mathcal{O}$-Kessar conjecture

Let $P$ be a finite $p$-group-group. Then there is $m=m(P) \in \mathbb{N}$ such that if $G$ is a finite group and $B$ is a block of $\mathcal{O} G$ with defect groups isomorphic to $P$, then ${\rm mf}_\mathcal{O}(B) \leq m$.

This conjecture is considered in an equivalent form in [EL18b] .

## Puig's conjecture

The following conjecture is very natural, but since it is very hard to reduce to quasisimple groups is known in very few cases.

### Puig's conjecture

Let $P$ be a finite $p$-group. Then there are only finitely many possible isomorphism classes source algebras for blocks of $kG$ for finite groups G with defect group isomorphic to $P$.

Here we mean an isomorphism of interior algebras.

## Broué's conjecture

This may be stated in several different forms. The basic version that we take here is:

### Broué's conjecture

Let $G$ be a finite group and $B$ be a block of $\mathcal{O} G$ with abelian defect group $D$. Let $B$ be the unique block of $\mathcal{O} N_G(D)$ with Brauer correspondent $B$. Then $B$ is derived equivalent to $b$.

It is expected that further there should be a splendid derived equivalence, and even that there should be a chain of perverse equivalences (see [CR13]).