C2xC2xC2 - Revision history

Charles Eaton at 16:49, 23 September 2019

2019-09-23T16:49:58Z

Charles Eaton at 22:50, 18 December 2018

2018-12-18T22:50:30Z

Charles Eaton: O-Picard groups

2018-10-26T06:55:08Z

O-Picard groups

Charles Eaton: Added references for derived equivalences

2018-10-05T15:47:02Z

Added references for derived equivalences

Charles Eaton: Removed under construction label

2018-10-05T10:41:58Z

Removed under construction label

Charles Eaton at 15:22, 26 September 2018

2018-09-26T15:22:58Z

Charles Eaton: Added reference

2018-09-23T17:16:38Z

Added reference

Charles Eaton at 08:42, 15 September 2018

2018-09-15T08:42:31Z

Charles Eaton: Added lifting column

2018-09-15T07:58:03Z

Added lifting column

Charles Eaton: Switched M(8,5,2) and M(8,5,3)

2018-09-08T15:46:49Z

Switched M(8,5,2) and M(8,5,3)

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>C_2 \times C_2 \times C_2</math> ==
-These were classified in [[References|[Ea16]]] using the [[Glossary#CFSG|CFSG]]. The Picard groups with respect to <math>\mathcal{O}</math> are computed in [[References#E|[EL18c]]].
+These were classified in [[References|[Ea16]]] using the [[Glossary#CFSG|CFSG]]. The Picard groups with respect to <math>\mathcal{O}</math> are computed in [[References#E|[EL18c]]] with the exception of the principal block of <math>J_1</math>, which has been computed by Eisele.
 {| class="wikitable"

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>C_2 \times C_2 \times C_2</math> ==
-These were classified in [[References|[Ea16]]] using the [[Glossary#CFSG|CFSG]]. The Picard groups with respect to <math>\mathcal{O}</math> are computed in [[References|[EL18c]]].
+These were classified in [[References|[Ea16]]] using the [[Glossary#CFSG|CFSG]]. The Picard groups with respect to <math>\mathcal{O}</math> are computed in [[References#E|[EL18c]]].
 {| class="wikitable"
@@ Line 37: / Line 37: @@
 |}
-[[M(8,5,2)]] and [[M(8,5,3)]] are derived equivalent over <math>\mathcal{O}</math>. This is a consequence of the derived equivalence between [[M(4,2,2)]] and [[M(4,2,3)]] (see [[References|[Ri96]]]).
+[[M(8,5,2)]] and [[M(8,5,3)]] are derived equivalent over <math>\mathcal{O}</math>. This is a consequence of the derived equivalence between [[M(4,2,2)]] and [[M(4,2,3)]] (see [[References#R|[Ri96]]]).
-[[M(8,5,4)]] and [[M(8,5,5)]] are derived equivalent over <math>\mathcal{O}</math>. See [[References|[Ro95]]].
+[[M(8,5,4)]] and [[M(8,5,5)]] are derived equivalent over <math>\mathcal{O}</math>. See [[References#R|[Ro95]]].
-[[M(8,5,6)]], [[M(8,5,7)]] and [[M(8,5,8)]] are derived equivalent over <math>\mathcal{O}</math>. See [[References|[Go97]]], [[References|[Ok97]]] and [[References|[CR13]]].
+[[M(8,5,6)]], [[M(8,5,7)]] and [[M(8,5,8)]] are derived equivalent over <math>\mathcal{O}</math>. See [[References#G|[Go97]]], [[References#O|[Ok97]]] and [[References#C|[CR13]]].

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>C_2 \times C_2 \times C_2</math> ==
-These were classified in [[References|[Ea16]]] using the [[Glossary#CFSG|CFSG]]. Each of the eight <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>.
+These were classified in [[References|[Ea16]]] using the [[Glossary#CFSG|CFSG]]. The Picard groups with respect to <math>\mathcal{O}</math> are computed in [[References|[EL18c]]].
 {| class="wikitable"
@@ Line 20: / Line 20: @@
 |-
-|[[M(8,5,1)]] || <math>k(C_2 \times C_2 \times C_2)</math> || 1 ||8 ||1 ||<math>1</math> || || ||1 ||1 ||
+|[[M(8,5,1)]] || <math>k(C_2 \times C_2 \times C_2)</math> || 1 ||8 ||1 ||<math>1</math> || <math>(C_2 \times C_2 \times C_2):GL_3(2)</math> || ||1 ||1 ||
 |-
-|[[M(8,5,2)]] || <math>B_0(k(A_5 \times C_2))</math> || 1 ||8 ||3 ||<math>C_3</math> || || ||1 ||1 ||
+|[[M(8,5,2)]] || <math>B_0(k(A_5 \times C_2))</math> || 1 ||8 ||3 ||<math>C_3</math> || <math>C_2 \times C_2</math>|| ||1 ||1 ||
 |-
-|[[M(8,5,3)]] || <math>k(A_4 \times C_2)</math> || 1 ||8 ||3 ||<math>C_3</math> || || ||1 ||1 ||
+|[[M(8,5,3)]] || <math>k(A_4 \times C_2)</math> || 1 ||8 ||3 ||<math>C_3</math> || <math>S_3 \times C_2</math>|| ||1 ||1 ||
 |-
-|[[M(8,5,4)]] || <math>k((C_2 \times C_2 \times C_2):C_7)</math> || 1 ||8 ||7 ||<math>C_7</math> || || ||1 ||1 ||
+|[[M(8,5,4)]] || <math>k((C_2 \times C_2 \times C_2):C_7)</math> || 1 ||8 ||7 ||<math>C_7</math> || <math>C_7:C_3</math>|| ||1 ||1 ||
 |-
-|[[M(8,5,5)]] || <math>B_0(kSL_2(8))</math> || 1 ||8 ||7 ||<math>C_7</math> || || ||1 ||1 ||
+|[[M(8,5,5)]] || <math>B_0(kSL_2(8))</math> || 1 ||8 ||7 ||<math>C_7</math> || <math>C_3</math> || ||1 ||1 ||
 |-
-|[[M(8,5,6)]] || <math>k((C_2 \times C_2 \times C_2):(C_7:C_3))</math> || 1 ||8 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||
+|[[M(8,5,6)]] || <math>k((C_2 \times C_2 \times C_2):(C_7:C_3))</math> || 1 ||8 ||5 ||<math>C_7:C_3</math> || <math>C_3</math> || ||1 ||1 ||
 |-
-|[[M(8,5,7)]] || <math>B_0(kJ_1)</math> || 1 ||8 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||
+|[[M(8,5,7)]] || <math>B_0(kJ_1)</math> || 1 ||8 ||5 ||<math>C_7:C_3</math> || <math>1</math> || ||1 ||1 ||
 |-
-|[[M(8,5,8)]] || <math>B_0(k{\rm Aut}(SL_2(8)))</math> || 1 ||8 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||
+|[[M(8,5,8)]] || <math>B_0(k{\rm Aut}(SL_2(8)))</math> || 1 ||8 ||5 ||<math>C_7:C_3</math> ||  <math>C_3</math> || ||1 ||1 ||
 |}

@@ Line 37: / Line 37: @@
 |}
-[[M(8,5,2)]] and [[M(8,5,3)]] are derived equivalent over <math>\mathcal{O}</math>.
+[[M(8,5,2)]] and [[M(8,5,3)]] are derived equivalent over <math>\mathcal{O}</math>. This is a consequence of the derived equivalence between [[M(4,2,2)]] and [[M(4,2,3)]] (see [[References|[Ri96]]]).
-[[M(8,5,4)]] and [[M(8,5,5)]] are derived equivalent over <math>\mathcal{O}</math>.
+[[M(8,5,4)]] and [[M(8,5,5)]] are derived equivalent over <math>\mathcal{O}</math>. See [[References|[Ro95]]].
-[[M(8,5,6)]], [[M(8,5,7)]] and [[M(8,5,8)]] are derived equivalent over <math>\mathcal{O}</math>.
+[[M(8,5,6)]], [[M(8,5,7)]] and [[M(8,5,8)]] are derived equivalent over <math>\mathcal{O}</math>. See [[References|[Go97]]], [[References|[Ok97]]] and [[References|[CR13]]].

← Older revision		Revision as of 10:41, 5 October 2018
Line 2:		Line 2:

	== Blocks with defect group <math>C_2 \times C_2 \times C_2</math> ==		== Blocks with defect group <math>C_2 \times C_2 \times C_2</math> ==
−
−	~~[[Image:under-construction.png\|50px\|left]]~~

	These were classified in [[References\|[Ea16]]] using the [[Glossary#CFSG\|CFSG]]. Each of the eight <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>.		These were classified in [[References\|[Ea16]]] using the [[Glossary#CFSG\|CFSG]]. Each of the eight <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>.

← Older revision		Revision as of 15:22, 26 September 2018
Line 2:		Line 2:

	== Blocks with defect group <math>C_2 \times C_2 \times C_2</math> ==		== Blocks with defect group <math>C_2 \times C_2 \times C_2</math> ==
		+
		+	[[Image:under-construction.png\|50px\|left]]

	These were classified in [[References\|[Ea16]]] using the [[Glossary#CFSG\|CFSG]]. Each of the eight <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>.		These were classified in [[References\|[Ea16]]] using the [[Glossary#CFSG\|CFSG]]. Each of the eight <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>.

@@ Line 9: / Line 9: @@
 ! scope="col"| Class
 ! scope="col"| Representative
-! scope="col"| # lifts / <math>\mathcal{O}</math>
+! scope="col"| [[Glossary|# lifts / <math>\mathcal{O}</math>]]
 ! scope="col"| <math>k(B)</math>
 ! scope="col"| <math>l(B)</math>

@@ Line 21: / Line 21: @@
 |[[M(8,5,1)]] || <math>k(C_2 \times C_2 \times C_2)</math> ||8 ||1 ||<math>1</math> || || ||1 ||1 ||
 |-
-|[[M(8,5,2)]] || <math>k(A_4 \times C_2)</math> ||8 ||3 ||<math>C_3</math> || || ||1 ||1 ||
+|[[M(8,5,2)]] || <math>B_0(k(A_5 \times C_2))</math> ||8 ||3 ||<math>C_3</math> || || ||1 ||1 ||
 |-
-|[[M(8,5,3)]] || <math>B_0(k(A_5 \times C_2))</math> ||8 ||3 ||<math>C_3</math> || || ||1 ||1 ||
+|[[M(8,5,3)]] || <math>k(A_4 \times C_2)</math> ||8 ||3 ||<math>C_3</math> || || ||1 ||1 ||
 |-
 |[[M(8,5,4)]] || <math>k((C_2 \times C_2 \times C_2):C_7)</math> ||8 ||7 ||<math>C_7</math> || || ||1 ||1 ||