Statements of conjectures - Revision history

Charles Eaton at 15:15, 21 August 2020

2020-08-21T15:15:47Z

Charles Eaton: /* Donovan's conjecture */ Link to Peter Donovan page

2020-08-21T15:05:13Z

‎Donovan's conjecture: Link to Peter Donovan page

Charles Eaton: /* Morita-Frobenius number conjectures */ Added commentary on Morita-Frobenius numbers

2020-06-29T08:13:50Z

‎Morita-Frobenius number conjectures: Added commentary on Morita-Frobenius numbers

Charles Eaton: /* \mathcal{O}-Kessar conjecture */

2019-08-02T16:55:08Z

‎\mathcal{O}-Kessar conjecture

Charles Eaton: /* Kessar's conjecture */

2019-08-02T16:54:33Z

‎Kessar's conjecture

Charles Eaton: /* Morita-Frobenius number conjectures */

2018-12-18T16:24:53Z

‎Morita-Frobenius number conjectures

Charles Eaton at 18:01, 13 October 2018

2018-10-13T18:01:39Z

Charles Eaton: Loewy and Ext conjectures

2018-10-13T17:59:55Z

Loewy and Ext conjectures

Charles Eaton at 14:43, 27 September 2018

2018-09-27T14:43:59Z

Charles Eaton at 15:39, 30 August 2018

2018-08-30T15:39:38Z

@@ Line 1: / Line 1: @@
 == Donovan's conjecture ==
-[[Image:Donovan.jpg|150px|thumb|right|[[Peter Donovan]] ]]
+[[Image:Donovan.jpg|150px|thumb|right|[[A few thoughts on the Donovan conjecture|Peter Donovan]] ]]
 Donovan's conjecture can be stated over a field or a complete discrete valuation ring. It is usual to assume the field is algebraically closed, and it is no loss to assume it is the algebraic closure <math>k</math> of the field with <math>p</math> elements, as in [[Notation|the notation section]].
@@ Line 10: / Line 10: @@
 </div>
-[[Peter Donovan|Here is a brief note about the conjecture by Peter Donovan]].
+[[A few thoughts on the Donovan conjecture|Here is a brief note about the conjecture by Peter Donovan]].
 The situation with choice of discrete valuation ring is a little more complicated, as discussed in [[Notation|the notation section]], and we take <math>\mathcal{O}</math> to be the ring of Witt vectors for <math>k</math>.

@@ Line 1: / Line 1: @@
 == Donovan's conjecture ==
 Donovan's conjecture can be stated over a field or a complete discrete valuation ring. It is usual to assume the field is algebraically closed, and it is no loss to assume it is the algebraic closure <math>k</math> of the field with <math>p</math> elements, as in [[Notation|the notation section]].
@@ Line 7: / Line 9: @@
 Let <math>P</math> be a finite <math>p</math>-group. Then there are only finitely many possible Morita equivalence classes for blocks of <math>kG</math> for finite groups G with defect group isomorphic to <math>P</math>.
 </div>
 The situation with choice of discrete valuation ring is a little more complicated, as discussed in [[Notation|the notation section]], and we take <math>\mathcal{O}</math> to be the ring of Witt vectors for <math>k</math>.

@@ Line 46: / Line 46: @@
 </div>
-This may be reformulated over <math>\mathcal{O}</math>. In [[References#E|[EEL18]]] it is shown that the <math>\mathcal{O}</math>-Donovan conjecture is equivalent to the Weak Donovan conjecture together with the following.
+In [[References#B|[BK07]]] Benson and Kessar produced examples where <math>{\rm mf}_k(B) = 2</math>. Livesey in [[References#L|[Liv19]]] showed that there are blocks with <math>{\rm mf}_\mathcal{O}(B)</math> arbitrarily large, and in [[References#E|[EiLiv20]]] Eisele and Livesey showed that <math>{\rm mf}_k(B)</math> can also be arbitrarily large. These however do not give counterexamples to Donovan's conjecture as examples giving large Morita-Frobenius numbers have large defect groups.
 <div class="boxed">

@@ Line 51: / Line 51: @@
 === <math>\mathcal{O}</math>-Kessar conjecture ===
-Let <math>P</math> be a finite <math>p</math>-group-group. Then there is $m=m(P) \in \mathbb{N}$ such that if <math>G</math> is a finite group and <math>B</math> is a block of <math>\mathcal{O} G</math> with defect groups isomorphic to <math>P</math>, then <math>{\rm mf}_\mathcal{O}(B) \leq m</math>.
+Let <math>P</math> be a finite <math>p</math>-group-group. Then there is <math>m=m(P) \in \mathbb{N}</math> such that if <math>G</math> is a finite group and <math>B</math> is a block of <math>\mathcal{O} G</math> with defect groups isomorphic to <math>P</math>, then <math>{\rm mf}_\mathcal{O}(B) \leq m</math>.
 </div>

@@ Line 43: / Line 43: @@
 <div class="boxed">
 === Kessar's conjecture ===
-Let <math>P</math> be a finite <math>p</math>-group-group. Then there is $m=m(P) \in \mathbb{N}$ such that if <math>G</math> is a finite group and <math>B</math> is a block of <math>kG</math> with defect groups isomorphic to <math>P</math>, then <math>{\rm mf}_k(B) \leq m</math>.
+Let <math>P</math> be a finite <math>p</math>-group-group. Then there is <math>m=m(P) \in \mathbb{N}</math> such that if <math>G</math> is a finite group and <math>B</math> is a block of <math>kG</math> with defect groups isomorphic to <math>P</math>, then <math>{\rm mf}_k(B) \leq m</math>.
 </div>
@@ Line 49: / Line 49: @@
 <div class="boxed">
 === <math>\mathcal{O}</math>-Kessar conjecture ===
 Let <math>P</math> be a finite <math>p</math>-group-group. Then there is $m=m(P) \in \mathbb{N}$ such that if <math>G</math> is a finite group and <math>B</math> is a block of <math>\mathcal{O} G</math> with defect groups isomorphic to <math>P</math>, then <math>{\rm mf}_\mathcal{O}(B) \leq m</math>.

@@ Line 39: / Line 39: @@
 == Morita-Frobenius number conjectures ==
-In [[References|[Ke05] ]] Kessar showed that the <math>k</math>-Donovan conjecture is equivalent to the Weak Donovan conjecture together with the following.
+In [[References#K|[Ke05] ]] Kessar showed that the <math>k</math>-Donovan conjecture is equivalent to the Weak Donovan conjecture together with the following.
 <div class="boxed">
@@ Line 46: / Line 46: @@
 </div>
-This may be reformulated over <math>\mathcal{O}</math>.
+This may be reformulated over <math>\mathcal{O}</math>. In [[References#E|[EEL18]]] it is shown that the <math>\mathcal{O}</math>-Donovan conjecture is equivalent to the Weak Donovan conjecture together with the following.
 <div class="boxed">
@@ Line 53: / Line 53: @@
 </div>
-This conjecture is considered in an equivalent form in [[References|[EL18b] ]].
+This conjecture is considered in an equivalent form in [[References#E|[EL18b] ]].
 == Puig's conjecture ==

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 <div class="boxed">
 === Ext space conjecture ===
-Let <math>P</math> be a finite <math>p</math>-group. Then there is <math>e=e(P) \in \mathbb{N}</math> such that for all blocks <math>B</math> of <math>kG</math> for finite groups G with defect group isomorphic to <math>P</math> and for all pairs <math>V, W</math> of simple <math>B</math>-modules, <math>\dim_k({\rm Ext}_{kG}(V,W)) \leq e</math>
+Let <math>P</math> be a finite <math>p</math>-group. Then there is <math>e=e(P) \in \mathbb{N}</math> such that for all blocks <math>B</math> of <math>kG</math> for finite groups G with defect group isomorphic to <math>P</math> and for all pairs <math>V, W</math> of simple <math>B</math>-modules, <math>\dim_k({\rm Ext}_{kG}^1(V,W)) \leq e</math>
 </div>

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 Let <math>P</math> be a finite <math>p</math>-group. Then there is <math>c=c(P) \in \mathbb{N}</math> such that for all blocks of <math>kG</math> for finite groups G with defect group isomorphic to <math>P</math>, the Cartan invariants are at most <math>c</math>.
 </div>
 == Morita-Frobenius number conjectures ==

← Older revision		Revision as of 14:43, 27 September 2018
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	</div>		</div>

−	It is expected that further there should be a splendid derived equivalence, and even that there should be a perverse ~~equivalence~~ (see [[References\|[CR13] ]]).	+	It is expected that further there should be a splendid derived equivalence, and even that there should be a chain of perverse equivalences (see [[References\|[CR13]]]).

← Older revision		Revision as of 15:39, 30 August 2018
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−	== Donovan's ~~Conjecture~~ ==	+	== Donovan's conjecture ==

	Donovan's conjecture can be stated over a field or a complete discrete valuation ring. It is usual to assume the field is algebraically closed, and it is no loss to assume it is the algebraic closure <math>k</math> of the field with <math>p</math> elements, as in [[Notation\|the notation section]].		Donovan's conjecture can be stated over a field or a complete discrete valuation ring. It is usual to assume the field is algebraically closed, and it is no loss to assume it is the algebraic closure <math>k</math> of the field with <math>p</math> elements, as in [[Notation\|the notation section]].