Difference between revisions of "Status of Donovan's conjecture"

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[[Image:Donovan.jpg|150px|thumb|right|Peter Donovan]]
 
[[Image:Donovan.jpg|150px|thumb|right|Peter Donovan]]
  
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In this page we list cases where Donovan's conjecture is known to hold. The first list is organised by <math>p</math>-group, and the second by class of finite groups.
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== Donovan's conjecture by <math>p</math>-group ==
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PAGE UNDER CONSTRUCTION!
 
  
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== Donovan's conjecture by <math>p</math>-group ==
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In the following, the column headed [[Statements of conjectures #Donovan's conjecture|Donovan's conjecture]] indicates whether the conjecture is known over <math>k</math> or <math>\mathcal{O}</math>.
  
 
{| class="wikitable"
 
{| class="wikitable"
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! scope="col"| Notes
 
! scope="col"| Notes
 
|-
 
|-
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|Cyclic <math>p</math>-groups || <math>\mathcal{O}</math> || Yes || ||
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|Cyclic <math>p</math>-groups || <math>\mathcal{O}</math> || Yes || [[References#L|[Li96]]] ||
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|-
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|<math>C_2 \times C_2</math> || <math>\mathcal{O}</math> || Yes || [[References#C|[CEKL11]]] || Donovan's conjecture without CFSG, Puig using CFSG
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|-
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|Abelian <math>2</math>-groups || <math>\mathcal{O}</math> || No || [[References#E|[EEL18]]] ||
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|-
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|Abelian <math>3</math>-groups || No || No || [[References#K|[Ko03]]] || Puig's conjecture known for principal blocks
 
|-
 
|-
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|<math>C_2 \times C_2</math> || <math>\mathcal{O}</math> || Yes || [CEKL11] || Donovan's conjecture without CFSG, Puig using CFSG
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|Dihedral <math>2</math>-groups || <math>k</math> || No || [[References#E|[Er87]]] ||
 
|-
 
|-
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|Abelian <math>2</math>-groups || <math>k</math> || No || ||
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|Semidihedral <math>2</math>-groups || <math>k</math> || No || [[References#E|[Er88c], [Er90b]]] ||
 
|-
 
|-
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|<math>C_3 \times C_3</math> || No || No || [Ko03] || Puig's conjecture known for principal blocks
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|<math>Q_8</math> || <math>\mathcal{O}</math> || No || [[References#E|[Er88a]]], [[References#E|[Er88b]]], [[References#K|[HKL07]]], [[References#E|[Ei16]]] ||
 
|-
 
|-
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|Dihedral <math>2</math>-groups || <math>k</math> || No || [Er90]] ||
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|<math>Q_8 \times C_{2^n}</math> || <math>\mathcal{O}</math> || No || [[References#E|[EL20]]] ||
 
|-
 
|-
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|Semidihedral <math>2</math>-groups || <math>k</math> || No || [Er90]] ||
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|<math>Q_8 \times Q_8</math> || <math>\mathcal{O}</math> || No || [[References#E|[EL20]]] ||
 
|-
 
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|Generalised quaternion <math>2</math>-groups || No || No || [Er90]] || Donovan's conjecture over <math>k</math> known if <math>l(B) \neq 2</math>
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|Generalised quaternion <math>2</math>-groups || No || No || [[References#E|[Er88a], [Er88b]]] || Donovan's conjecture over <math>\mathcal{O}</math> known if <math>l(B) \neq 2</math><ref>When <math>l(B) \neq 2</math>, each <math>k</math>-Morita equivalence class lifts uniquely to <math>\mathcal{O}</math> by [[References|[Ei16]]].</ref>
 
|-  
 
|-  
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|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 || [EKS12] || Additional assumptions on <math>\mathcal{O}</math>, which may not be necessary
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|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 || [[References#E|[EKS12]]] ||
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|-
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|Metacyclic <math>2</math>-groups of nonmaximal class || <math>\mathcal{O}</math> || No || [[References#C|[CG12]]], [[References#S|[Sa12b]]] || All blocks nilpotent
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|}
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== Donovan's conjecture by class of group or block ==
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In the table, the column headed Donovan's conjecture indicates whether the conjecture is known over <math>k</math> or <math>\mathcal{O}</math>.
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Note that knowing the <math>\mathcal{O}</math>-Donovan conjecture or [[Statements of conjectures #Puig's conjecture|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.
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{| class="wikitable"
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|-
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! scope="col"| Groups
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! scope="col"| Blocks
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! scope="col"| Donovan's conjecture
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! scope="col"| Puig's conjecture
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! scope="col"| References
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! scope="col"| Notes
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|-
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|<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#L|[Li18d,10.6.2]]]
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|-
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|Symmetric groups || All || <math>\mathcal{O}</math> || Yes || Over <math>k</math> by [[References#S|[Sc91]]], Puig's conjecture by [[References#P|[Pu94]]] ||
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|-
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|Double covers of symmetric groups || All || <math>\mathcal{O}</math> || Yes || [[References#K|[Ke96]]] ||
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|-
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|Alternating groups and their double covers || All || <math>\mathcal{O}</math> || Yes || [[References#K|[Ke02], [Ke96]]] ||
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|-
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|<math>GL_n(q)</math> for fixed <math>q</math> || Unipotent blocks || <math>\mathcal{O}</math> || Yes || Over <math>k</math> by [[References#J|[Jo96]]], Puig's conjecture by [[References#K|[Ke01]]] ||
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|-
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|Classical groups || Unipotent blocks for linear primes || <math>\mathcal{O}</math> || Yes || [[References#H|[HK00], [HK05]]] || Detailed results beyond those stated here
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|-
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|Weyl groups of type <math>B, D</math> || All || <math>\mathcal{O}</math> || Yes || [[References#K|[Ke00]]] ||
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|-
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|Arbitrary groups || Blocks with [[Glossary#Trivial intersection subgroup|trivial intersection]] defect groups || <math>\mathcal{O}</math> || No || [[References#A|[AE04]]] ||
 
|}
 
|}
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== Weak Donovan conjecture ==
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As described in [[References #D|[Dü04]]] the [[Statements of conjectures #Weak Donovans conjecture|Weak Donovan conjecture]] is equivalent to [[Statements of conjectures #Weak Donovans conjecture|bounding the dimensions of the Ext spaces between simple modules]] and [[Statements of conjectures #Weak Donovans conjecture|bounding the Loewy length]]. See [[References #G|[GT19]]] and [[References #S|[Sh20]]] for progress on the former problem.
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== Notes ==
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<references />

Revision as of 08:59, 29 June 2020

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]
[math]Q_8 \times C_{2^n}[/math] [math]\mathcal{O}[/math] No [EL20]
[math]Q_8 \times Q_8[/math] [math]\mathcal{O}[/math] No [EL20]
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] and [Sh20] for progress on the former problem.

Notes

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