# Statements of conjectures

## Contents

## 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 [math]k[/math] of the field with [math]p[/math] elements, as in the notation section.

### [math]k[/math]-Donovan conjecture

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

Here is a brief note about the conjecture by Peter Donovan.

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

### [math]\mathcal{O}[/math]-Donovan conjecture

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

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

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

### Loewy length conjecture

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

### Ext space conjecture

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

## Morita-Frobenius number conjectures

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

### Kessar's conjecture

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

In [BK07] Benson and Kessar produced examples where [math]{\rm mf}_k(B) = 2[/math]. Livesey in [Liv19] showed that there are blocks with [math]{\rm mf}_\mathcal{O}(B)[/math] arbitrarily large, and in [EiLiv20] Eisele and Livesey showed that [math]{\rm mf}_k(B)[/math] can also be arbitrarily large. These however do not give counterexamples to Donovan's conjecture as examples giving large Morita-Frobenius numbers have large defect groups.

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

### [math]\mathcal{O}[/math]-Kessar conjecture

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

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

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

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]).