View source for Status of Donovan's conjecture

[[Image:Donovan.jpg|150px|thumb|right|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>.

{| class="wikitable"
|-
! scope="col"| <math>p</math>-groups
! scope="col"| Donovan's conjecture
! scope="col"| Puig's conjecture
! scope="col"| References
! scope="col"| Notes
|-
|Cyclic <math>p</math>-groups || <math>\mathcal{O}</math> || Yes || [[References|[Li96]]] ||
|-
|<math>C_2 \times C_2</math> || <math>\mathcal{O}</math> || Yes || [[References|[CEKL11]]] || Donovan's conjecture without CFSG, Puig using CFSG
|-
|Abelian <math>2</math>-groups || <math>\mathcal{O}</math> || No || [[References|[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 || [[References|[Er87]]] ||
|-
|Semidihedral <math>2</math>-groups || <math>k</math> || No || [[References|[Er88c], [Er90b]]] ||
|-
|<math>Q_8</math> || <math>\mathcal{O}</math> || No || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] ||
|-
|Generalised quaternion <math>2</math>-groups || No || No || [[References|[Er88a], [Er88b]]] || Donovan's conjecture over <math>k</math> known if <math>l(B) \neq 2</math>
|- 
|Minimal nonabelian <math>2</math>-groups <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 || [[References|[EKS12]]] || 
|-
|Metacyclic <math>2</math>-groups of nonmaximal class || <math>\mathcal{O}</math> || No || [[References|[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.

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