Status of Donovan's conjecture
Contents
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][1] |
| 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] for progress on the former problem.
