(C2)^5 - Revision history

Charles Eaton: "To do" marker

2022-05-17T09:18:39Z

"To do" marker

CesareGArdito: /* Blocks with defect group (C_2)^5 */

2020-11-13T09:40:21Z

‎Blocks with defect group (C_2)^5

CesareGArdito at 09:40, 13 November 2020

2020-11-13T09:40:08Z

CesareGArdito: /* Blocks with defect group (C_2)^5 */

2020-11-13T09:39:25Z

‎Blocks with defect group (C_2)^5

CesareGArdito at 15:52, 9 December 2019

2019-12-09T15:52:26Z

CesareGArdito at 15:50, 9 December 2019

2019-12-09T15:50:07Z

CesareGArdito: /* Blocks with defect group (C_2)^5 */

2019-12-09T15:48:50Z

‎Blocks with defect group (C_2)^5

CesareGArdito at 15:45, 9 December 2019

2019-12-09T15:45:56Z

CesareGArdito: /* Blocks with defect group (C_2)^5 */

2019-12-09T15:44:16Z

‎Blocks with defect group (C_2)^5

CesareGArdito at 15:00, 9 December 2019

2019-12-09T15:00:29Z

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>(C_2)^5</math> ==
-These were classified in [[References#A|[Ar19]]] using the [[Glossary#CFSG|CFSG]]. Each of the 34 <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The known Picard groups were computed in [[References#E|[EL18c]]] or using the main theorem of [[References#L|[Liv19]]].
+These were classified in [[References#A|[Ar19]]] using the [[Glossary#CFSG|CFSG]]. Each of the 34 <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The known Picard groups were computed in [[References#E|[EL18c]]] or using the main theorem of [[References#L|[Liv19]]]. TO DO: PICARD GROUPS COMPUTED AND FILLED IN WHERE METHOD KNOWN
 {| class="wikitable"

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>(C_2)^5</math> ==
-These were classified in [[References#A|[Ar19]]] using the [[Glossary#CFSG|CFSG]]. Each of the 34 <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The known Picard groups were computed in [[References#E|[EL18c]]] or using the main theorem of [[References#L|[Liv19]]]
+These were classified in [[References#A|[Ar19]]] using the [[Glossary#CFSG|CFSG]]. Each of the 34 <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The known Picard groups were computed in [[References#E|[EL18c]]] or using the main theorem of [[References#L|[Liv19]]].
 {| class="wikitable"

@@ Line 92: / Line 92: @@
 If a block of a finite group is Morita equivalent to another block with defect group <math>(C_2)^5</math>, then it also has defect group <math>(C_2)^5</math>.
-Blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group), as shown in [[References#A|[Ar19]]] and [[References#A|[AS20]]]
+Blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group), as shown in [[References#A|[Ar19]]] and [[References#A|[AS20]]].
 All blocks with defect group  <math>(C_2)^5</math> are derived equivalent to their Brauer correspondent, and each derived equivalence class is determined by the number of simple modules and the inertial quotient (and its action on the defect group) [[References#A|[AS20]]].
 It is unknown whether these derived equivalences are splendid, because we cannot say anything about the sources of the Morita equivalences determined in [[References#A|[Ar19]]].

@@ Line 28: / Line 28: @@
 |[[M(32,51,4)]] || <math>k(((C_2)^4 :C_3) \times C_2)</math> || 1 ||16 ||3 ||<math>C_3</math> || || ||1 ||1 || The action of <math>C_3</math> comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math>
 |-
-|[[M(32,51,5)]] || <math>k(((C_2)^4 :C_5) \times C_2)</math> || 1 ||16 ||5 ||<math>C_5</math>, ? || || ||1 ||1 ||
+|[[M(32,51,5)]] || <math>k(((C_2)^4 :C_5) \times C_2)</math> || 1 ||16 ||5 ||<math>C_5</math> || || ||1 ||1 ||
 |-
 |[[M(32,51,6)]] || <math>k(((C_2)^3:C_7) \times (C_2)^2)</math> || 1 ||32 ||7 ||<math>C_7</math> || || ||1 ||1 ||
@@ Line 40: / Line 40: @@
 |[[M(32,51,10)]] || <math>B_0(k(A_5 \times A_5 \times C_2))</math> || 1 ||32 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||
 |-
-|[[M(32,51,11)]] || <math>k((C_2)^4 : C_{15} \times C_2)</math> || 1 ||32 ||15 ||<math>C_{15}</math>, ? || || ||1 ||1 ||
+|[[M(32,51,11)]] || <math>k((C_2)^4 : C_{15} \times C_2)</math> || 1 ||32 ||15 ||<math>C_{15}</math> || || ||1 ||1 ||
 |-
-|[[M(32,51,12)]] || <math>B_0(k(SL_2(16) \times C_2))</math> || 1 ||32 ||15 ||<math>C_{15}</math>, ? || || ||1 ||1 ||
+|[[M(32,51,12)]] || <math>B_0(k(SL_2(16) \times C_2))</math> || 1 ||32 ||15 ||<math>C_{15}</math> || || ||1 ||1 ||
 |-
 |[[M(32,51,13)]] || <math>k(((C_2)^3:C_7) \times A_4)</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 ||
@@ Line 92: / Line 92: @@
 If a block of a finite group is Morita equivalent to another block with defect group <math>(C_2)^5</math>, then it also has defect group <math>(C_2)^5</math>.
-With the possible exceptions of [[M(32,51,5)]], [[M(32,51,11)]] and [[M(32,51,12)]], blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group).
+Blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group), as shown in [[References#A|[Ar19]]] and [[References#A|[AS20]]]
-All blocks with defect group  <math>(C_2)^5</math> are splendid derived equivalent to their Brauer correspondent, with the possible exceptions of:
+All blocks with defect group  <math>(C_2)^5</math> are derived equivalent to their Brauer correspondent, and each derived equivalence class is determined by the number of simple modules and the inertial quotient (and its action on the defect group) [[References#A|[AS20]]].
-All the derived equivalences above are splendid, and also occur over <math>\mathcal{O}</math>.
+It is unknown whether these derived equivalences are splendid, because we cannot say anything about the sources of the Morita equivalences determined in [[References#A|[Ar19]]].

@@ Line 96: / Line 96: @@
 All blocks with defect group  <math>(C_2)^5</math> are splendid derived equivalent to their Brauer correspondent, with the possible exceptions of:
 * Blocks in [[M(32,51,34)]].
-* Possible blocks in [[M(32,51,5)]] with inertial quotient <math>C_7:C_3</math> (acting as in [[M(32,51,20]]).
+* Possible blocks in [[M(32,51,5)]] with inertial quotient <math>C_7:C_3</math> (acting as in [[M(32,51,20]])).
 * Possible blocks in [[M(32,51,11)]] or [[M(32,51,12)]] with inertial quotient <math>(C_7:C_3) \times C_3</math>.

@@ Line 94: / Line 94: @@
 With the possible exceptions of [[M(32,51,5)]], [[M(32,51,11)]] and [[M(32,51,12)]], blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group).
-With the possible exceptions of blocks in [[M(32,51,34)]], possible blocks in [[M(32,51,5)]] with inertial quotient <math>C_7:C_3</math> and possible blocks in [[M(32,51,11)]] or [[M(32,51,12)]] with inertial quotient <math>C_7:C_3 \times C_3</math>, all blocks are splendid derived equivalent to their Brauer correspondent.
+All blocks with defect group  <math>(C_2)^5</math> are splendid derived equivalent to their Brauer correspondent, with the possible exceptions of:
 At the moment it is unknown whether [[M(32,51,5)]] contains blocks with inertial quotient <math>C_7:C_3</math> (as in [[M(32,51,20)]]), and whether [[M(32,51,11)]] and [[M(32,51,12)]] contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math>.
-All the derived equivalences above also occur over <math>\mathcal{O}</math>.
+All the derived equivalences above are splendid, and also occur over <math>\mathcal{O}</math>.