# Status of Donovan's conjecture

Peter Donovan

## Donovan's conjecture by $p$-group

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

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

Note that knowing the $\mathcal{O}$-Donovan conjecture or Puig's conjecture for blocks for a class of groups does not necessarily mean that the $\mathcal{O}$-lifts or source algebras of the $k$-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
$p$-solvable groups All $\mathcal{O}$ Yes Over $k$ by [Ku81], Puig's conjecture by [Pu09] See [Li18d,10.6.2]
Symmetric groups All $\mathcal{O}$ Yes Over $k$ by [Sc91], Puig's conjecture by [Pu94]
Double covers of symmetric groups All $\mathcal{O}$ Yes [Ke96]
Alternating groups and their double covers All $\mathcal{O}$ Yes [Ke02], [Ke96]
$GL_n(q)$ for fixed $q$ Unipotent blocks $\mathcal{O}$ Yes Over $k$ by [Jo96], Puig's conjecture by [Ke01]
Classical groups Unipotent blocks for linear primes $\mathcal{O}$ Yes [HK00], [HK05] Detailed results beyond those stated here
Weyl groups of type $B, D$ All $\mathcal{O}$ Yes [Ke00]
Arbitrary groups Blocks with trivial intersection defect groups $\mathcal{O}$ No [AE04]

## Notes

1. When $l(B) \neq 2$, each $k$-Morita equivalence class lifts uniquely to $\mathcal{O}$ by [Ei16].