Status of Donovan's conjecture

Peter Donovan

Donovan's conjecture by [math]p[/math]-group

In the following, the column headed Donovan's conjecture indicates whether the conjecture is known over [math]k[/math] or [math]\mathcal{O}[/math].

[math]p[/math]-groups	Donovan's conjecture	Puig's conjecture	References	Notes
Cyclic [math]p[/math]-groups	[math]\mathcal{O}[/math]	Yes	[Li96]
[math]C_2 \times C_2[/math]	[math]\mathcal{O}[/math]	Yes	[CEKL11]	Donovan's conjecture without CFSG, Puig using CFSG
Abelian [math]2[/math]-groups	[math]\mathcal{O}[/math]	No	[EEL18]
Abelian [math]3[/math]-groups	No	No	[Ko03]	Puig's conjecture known for principal blocks
Dihedral [math]2[/math]-groups	[math]k[/math]	No	[Er87]
Semidihedral [math]2[/math]-groups	[math]k[/math]	No	[Er88c], [Er90b]
[math]Q_8[/math]	[math]\mathcal{O}[/math]	No	[Er88a], [Er88b], [HKL07], [Ei16]
Generalised quaternion [math]2[/math]-groups	No	No	[Er88a], [Er88b]	Donovan's conjecture over [math]\mathcal{O}[/math] known if [math]l(B) \neq 2[/math]
Minimal nonabelian [math]2[/math]-groups of the form [math]\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle[/math]	[math]\mathcal{O}[/math]	No	[EKS12]
Metacyclic [math]2[/math]-groups of nonmaximal class	[math]\mathcal{O}[/math]	No	[CG12], [Sa12b]	All blocks nilpotent

Donovan's conjecture by class of group or block

In the table, the column headed Donovan's conjecture indicates whether the conjecture is known over [math]k[/math] or [math]\mathcal{O}[/math].

Note that knowing the [math]\mathcal{O}[/math]-Donovan conjecture or Puig's conjecture for blocks for a class of groups does not necessarily mean that the [math]\mathcal{O}[/math]-lifts or source algebras of the [math]k[/math]-Morita equivalence classes involved are known. This is only known for elements of the Morita equivalence class which occur as blocks of groups in that class.

Groups	Blocks	Donovan's conjecture	Puig's conjecture	References	Notes
[math]p[/math]-solvable groups	All	[math]\mathcal{O}[/math]	Yes	Over [math]k[/math] by [Ku81], Puig's conjecture by [Pu09]	See [Li18d,10.6.2]
Symmetric groups	All	[math]\mathcal{O}[/math]	Yes	Over [math]k[/math] by [Sc91], Puig's conjecture by [Pu94]
Double covers of symmetric groups	All	[math]\mathcal{O}[/math]	Yes	[Ke96]
Alternating groups and their double covers	All	[math]\mathcal{O}[/math]	Yes	[Ke02], [Ke96]
[math]GL_n(q)[/math] for fixed [math]q[/math]	Unipotent blocks	[math]\mathcal{O}[/math]	Yes	Over [math]k[/math] by [Jo96], Puig's conjecture by [Ke01]
Classical groups	Unipotent blocks for linear primes	[math]\mathcal{O}[/math]	Yes	[HK00], [HK05]	Detailed results beyond those stated here
Weyl groups of type [math]B, D[/math]	All	[math]\mathcal{O}[/math]	Yes	[Ke00]
Arbitrary groups	Blocks with trivial intersection defect groups	[math]\mathcal{O}[/math]	No	[AE04]

Weak Donovan conjecture

As described in [Dü04] the Weak Donovan conjecture is equivalent to bounding the dimensions of the Ext spaces between simple modules and bounding the Loewy length. See [GT19] and [Sh20] for progress on the former problem.