C3xC3 - Revision history

Charles Eaton: Some Picard groups

2022-05-24T09:45:28Z

Some Picard groups

Charles Eaton at 10:04, 11 April 2019

2019-04-11T10:04:52Z

Charles Eaton: Added new class M(9,2,24), renaming subsequent class

2019-04-10T10:07:58Z

Added new class M(9,2,24), renaming subsequent class

Charles Eaton at 14:01, 5 April 2019

2019-04-05T14:01:11Z

Charles Eaton at 08:50, 11 March 2019

2019-03-11T08:50:18Z

Charles Eaton at 15:21, 8 March 2019

2019-03-08T15:21:53Z

Charles Eaton at 15:18, 8 March 2019

2019-03-08T15:18:43Z

Charles Eaton at 15:00, 8 March 2019

2019-03-08T15:00:25Z

Charles Eaton at 10:07, 18 January 2019

2019-01-18T10:07:35Z

Charles Eaton: M(9,2,25)

2018-12-20T11:49:52Z

M(9,2,25)

@@ Line 5: / Line 5: @@
 [[Image:under-construction.png|50px|left]]
-Source algebra equivalence classes of principal blocks with this defect group have been classified by Koshitani in [[References|[Ko03]]] using the [[Glossary#CFSG|CFSG]]. This accounts for classes [[M(9,2,1)]] to [[M(9,2,22)]]. Note that it does not follow from the results in [[References|[Ko03]]] that there is a unique <math>\mathcal{O}</math>-Morita equivalence class for each of these <math>k</math>-Morita equivalence classes as they may also contain non-principal blocks.
+Source algebra equivalence classes of principal blocks with this defect group have been classified by Koshitani in [[References|[Ko03]]] using the [[Glossary#CFSG|CFSG]]. This accounts for classes [[M(9,2,1)]] to [[M(9,2,22)]]. Note that it does not follow from the results in [[References#K|[Ko03]]] that there is a unique <math>\mathcal{O}</math>-Morita equivalence class for each of these <math>k</math>-Morita equivalence classes as they may also contain non-principal blocks. Some Picard groups calculated in [[References#M|[Mar]]].
 The numerical invariants of arbitrary blocks with defect group <math>C_3 \times C_3</math> were calculated for all [[Glossary#Inertial quotient|inertial quotients]] except <math>C_8</math> and <math>Q_8</math> by Kiyota in [[References|[Ki84]]], and these cases remain an open problem. Kiyota's calculations do not involve the [[Glossary#CFSG|CFSG]].
@@ Line 28: / Line 28: @@
 |[[M(9,2,1)]] || <math>k(C_3 \times C_3)</math> || 1 ||9 ||1 ||<math>1</math> || <math>(C_3 \times C_3):GL_2(3)</math> || ||1 ||1 ||
 |-
-|[[M(9,2,2)]] ||  <math>k(S_3 \times C_3)</math> ||  ||9 ||2 ||<math>C_2</math> || || ||1 ||1 ||
+|[[M(9,2,2)]] ||  <math>k(S_3 \times C_3)</math> ||  ||9 ||2 ||<math>C_2</math> || <math>C_2 \times S_3</math> || ||1 ||1 ||
 |-
-|[[M(9,2,3)]] ||  <math>k(C_3 \times C_3):C_2</math> ||  ||6 ||2 ||<math>C_2</math> (power of Singer cycle) || || ||1 ||1 || SmallGroup(18,4)
+|[[M(9,2,3)]] ||  <math>k(C_3 \times C_3):C_2</math> ||  ||6 ||2 ||<math>C_2</math> (power of Singer cycle) || <math>C_8</math> || ||1 ||1 || SmallGroup(18,4)
 |-
-|[[M(9,2,4)]] ||  <math>k(S_3 \times S_3)</math> ||  ||9 ||4 ||<math>C_2 \times C_2</math> || || ||1 ||1 ||
+|[[M(9,2,4)]] ||  <math>k(S_3 \times S_3)</math> ||  ||9 ||4 ||<math>C_2 \times C_2</math> || <math>D_8</math> || ||1 ||1 ||
 |-
-|[[M(9,2,5)]] ||  <math>k(C_3 \times C_3):C_4</math> ||  ||6 ||4 ||<math>C_4</math> || || ||1 ||1 ||
+|[[M(9,2,5)]] ||  <math>k(C_3 \times C_3):C_4</math> ||  ||6 ||4 ||<math>C_4</math> || <math>C_2 \times D_8</math> || ||1 ||1 ||
 |-
-|[[M(9,2,6)]] ||  <math>B_0(kA_6)</math> ||  ||6 ||4 ||<math>C_4</math> || || ||1 ||1 ||
+|[[M(9,2,6)]] ||  <math>B_0(kA_6)</math> ||  ||6 ||4 ||<math>C_4</math> || <math>C_2 \times C_2</math> || ||1 ||1 ||
 |-
-|[[M(9,2,7)]] ||  <math>B_0(kA_7)</math> ||  ||6 ||4 ||<math>C_4</math> || || ||1 ||1 ||
+|[[M(9,2,7)]] ||  <math>B_0(kA_7)</math> ||  ||6 ||4 ||<math>C_4</math> || <math>C_2 \times C_2</math> || ||1 ||1 ||
 |-
-|[[M(9,2,8)]] ||  <math>k(C_3 \times C_3):C_8</math> ||  ||9 ||8 ||<math>C_8</math> || || ||1 ||1 ||
+|[[M(9,2,8)]] ||  <math>k(C_3 \times C_3):C_8</math> ||  ||9 ||8 ||<math>C_8</math> || <math>S_4</math> || ||1 ||1 ||
 |-
-|[[M(9,2,9)]] ||  <math>B_0(kPGL_2(9))</math> ||  ||9 ||8 ||<math>C_8</math> || || ||1 ||1 ||
+|[[M(9,2,9)]] ||  <math>B_0(kPGL_2(9))</math> ||  ||9 ||8 ||<math>C_8</math> || <math>C_2 \times C_2</math> || ||1 ||1 ||
 |-
-|[[M(9,2,10)]] ||  <math>k(C_3 \times C_3):D_8</math> ||  ||9 ||5 ||<math>D_8</math> || || ||1 ||1 ||
+|[[M(9,2,10)]] ||  <math>k(C_3 \times C_3):D_8</math> ||  ||9 ||5 ||<math>D_8</math> || <math>D_8</math> || ||1 ||1 ||
 |-
-|[[M(9,2,11)]] ||  <math>B_0(kA_8)</math> ||  ||9 ||5 ||<math>D_8</math> || || ||1 ||1 ||
+|[[M(9,2,11)]] ||  <math>B_0(kA_8)</math> ||  ||9 ||5 ||<math>D_8</math> || <math>C_2</math> || ||1 ||1 ||
 |-
-|[[M(9,2,12)]] ||  <math>B_0(S_6)</math> ||  ||9 ||5 ||<math>D_8</math> || || ||1 ||1 ||
+|[[M(9,2,12)]] ||  <math>B_0(S_6)</math> ||  ||9 ||5 ||<math>D_8</math> || <math>C_2 \times C_2</math> || ||1 ||1 ||
 |-
-|[[M(9,2,13)]] ||  <math>B_0(S_7)</math> ||  ||9 ||5 ||<math>D_8</math> || || ||1 ||1 ||
+|[[M(9,2,13)]] ||  <math>B_0(S_7)</math> ||  ||9 ||5 ||<math>D_8</math> || <math>C_2 \times C_2</math> || ||1 ||1 ||
 |-
-|[[M(9,2,14)]] ||  <math>k(C_3 \times C_3):Q_8</math> ||  ||6 ||5 ||<math>Q_8</math> || || ||1 ||1 ||
+|[[M(9,2,14)]] ||  <math>k(C_3 \times C_3):Q_8</math> ||  ||6 ||5 ||<math>Q_8</math> || <math>S_4</math> || ||1 ||1 ||
 |-
-|[[M(9,2,15)]] ||  <math>B_0(kM_{22})</math> ||  ||6 ||5 ||<math>Q_8</math> || || ||1 ||1 ||
+|[[M(9,2,15)]] ||  <math>B_0(kM_{22})</math> ||  ||6 ||5 || <math>Q_8</math> || <math>C_2 \times C_2</math> || ||1 ||1 || Also <math>M_{10}</math>
 |-
 |[[M(9,2,16)]] ||  <math>B_0(kPSL_3(4))</math> ||  ||6 ||5 ||<math>Q_8</math> || || ||1 ||1 ||
 |-
-|[[M(9,2,17)]] ||  <math>k(C_3 \times C_3):SD_{16}</math> ||  ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 ||
+|[[M(9,2,17)]] ||  <math>k(C_3 \times C_3):SD_{16}</math> ||  ||9 ||7 ||<math>SD_{16}</math> || <math>C_2 \times C_2</math> || ||1 ||1 ||
 |-
 |[[M(9,2,18)]] ||  <math>B_0(kM_{11})</math> ||  ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 ||
@@ Line 64: / Line 64: @@
 |[[M(9,2,19)]] ||  <math>B_0(kHS)</math> ||  ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 ||
 |-
-|[[M(9,2,20)]] ||  <math>B_0(kM_{23})</math> ||  ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 ||
+|[[M(9,2,20)]] ||  <math>B_0(kM_{23})</math> ||  ||9 ||7 ||<math>SD_{16}</math> || <math>1</math> || ||1 ||1 ||
 |-
 |[[M(9,2,21)]] ||  <math>B_0(kPSL_3(4).2_3)</math> ||  ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 ||  Extension by graph automorphism
 |-
-|[[M(9,2,22)]] ||  <math>B_0(k{\rm Aut}(S_6))</math> ||  ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 ||
+|[[M(9,2,22)]] ||  <math>B_0(k{\rm Aut}(S_6))</math> ||  ||9 ||7 ||<math>SD_{16}</math> || <math>C_2 \times C_2</math> || ||1 ||1 ||
 |-
 |[[M(9,2,23)]] || Faithful block of <math>k((C_3 \times C_3):Q_8)</math>, in which <math>Z(Q_8)</math> acts trivially ||  || 6 || 1 || <math>C_2 \times C_2</math> || || || 1 || 1 || SmallGroup(72,24)

@@ Line 7: / Line 7: @@
 Source algebra equivalence classes of principal blocks with this defect group have been classified by Koshitani in [[References|[Ko03]]] using the [[Glossary#CFSG|CFSG]]. This accounts for classes [[M(9,2,1)]] to [[M(9,2,22)]]. Note that it does not follow from the results in [[References|[Ko03]]] that there is a unique <math>\mathcal{O}</math>-Morita equivalence class for each of these <math>k</math>-Morita equivalence classes as they may also contain non-principal blocks.
-The numerical invariants of arbitrary blocks with defect group <math>C_3 \times C_3</math> were calculated for all [[Glossary#Inertial quotient|inertial quotients]] except <math>C_8</math> and <math>Q_8</math> by Kiyota in [[References|[Ki84]]], and these cases remain an open problem. These calculations do not involve the [[Glossary#CFSG|CFSG]].
+The numerical invariants of arbitrary blocks with defect group <math>C_3 \times C_3</math> were calculated for all [[Glossary#Inertial quotient|inertial quotients]] except <math>C_8</math> and <math>Q_8</math> by Kiyota in [[References|[Ki84]]], and these cases remain an open problem. Kiyota's calculations do not involve the [[Glossary#CFSG|CFSG]].
 '''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''

@@ Line 72: / Line 72: @@
 |[[M(9,2,23)]] || Faithful block of <math>k((C_3 \times C_3):Q_8)</math>, in which <math>Z(Q_8)</math> acts trivially ||  || 6 || 1 || <math>C_2 \times C_2</math> || || || 1 || 1 || SmallGroup(72,24)
 |-
-|[[M(9,2,24)]] || <math>B_{10}(k(4.M_{22}))</math> ||  ||6 ||5 ||<math>Q_8</math> || || ||1 ||1 ||
+|[[M(9,2,24)]] || Faithful block of <math>k((C_3 \times C_3):SD_{16})</math>, in which <math>Z(SD_{16})</math> acts trivially ||  || 6 || 2 || <math>D_8</math> || || || 1 || 1 ||
 |-
-|[[M(9,2,25)]] || <math>B_{7}(k(2.HS))</math> ||  ||9 ||5 ||<math>D_8</math> || || ||1 ||1 ||
+|[[M(9,2,25)]] || <math>B_{10}(k(4.M_{22}))</math> ||  ||6 ||5 ||<math>Q_8</math> || || ||1 ||1 ||
 |-
-|[[M(9,2,26)]] || <math>B_{2}(k(HS))</math> ||  ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 ||
+|[[M(9,2,26)]] || <math>B_{7}(k(2.HS))</math> ||  ||9 ||5 ||<math>D_8</math> || || ||1 ||1 ||
 |-
 | || <math>B_3(kCo_1)</math> || || 9 || 5 || <math>D_8</math> || || || 1 || 1 ||

@@ Line 86: / Line 86: @@
 | || Block of <math>kFi_{24}'.2</math> covering <math>B_2(kFi_{24}')</math> || || 9 || 5 || <math>D_8</math> || || || 1 || 1 ||
 |-
-| || <math>B_2(k(HS.2))</math> || || 9 || 7 || <math>SD_{16}</math> || || || 1 || 1 ||
+| || <math>B_3(k(HS.2))</math> || || 9 || 5 || <math>D_8</math> || || || 1 || 1 ||
 |-
 | || <math>B_6(k(2.M_{22}))</math> || || 6 || 5 || || || || 1 || 1 ||
@@ Line 101: / Line 103: @@
 Some open problems:
 *Determine whether <math>B_6(k(2.M_{22}))</math> is Morita equivalent to <math>(C_3 \times C_3):Q_8</math>.
 *Determine whether <math>B_3(kCo_1), B_6(kJ_4), B_2(kFi_{24}'), B_2(kB), B_3(kB), B_6(kB)</math> are in Morita equivalence classes not listed in the table.

@@ Line 77: / Line 77: @@
 |-
 |[[M(9,2,26)]] || <math>B_{2}(k(HS))</math> ||  ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 ||
 |}

← Older revision		Revision as of 15:18, 8 March 2019
Line 78:		Line 78:
	\|[[M(9,2,26)]] \|\| <math>B_{2}(k(HS))</math> \|\| \|\|9 \|\|7 \|\|<math>SD_{16}</math> \|\| \|\| \|\|1 \|\|1 \|\|		\|[[M(9,2,26)]] \|\| <math>B_{2}(k(HS))</math> \|\| \|\|9 \|\|7 \|\|<math>SD_{16}</math> \|\| \|\| \|\|1 \|\|1 \|\|
	\|}		\|}
		+
		+	Some open problems:
		+	*Determine whether <math>B_6(k(2.M_{22}))</math> is Morita equivalent to <math>(C_3 \times C_3):Q_8</math>.
		+	*Determine whether <math>B_2(k(HN))</math> is Morita equivalent to <math>B_0(k(He))</math>.
		+	*Determine whether <math>B_3(kCo_1), B_6(kJ_4), B_2(kFi_{24}'), B_2(kB), B_3(kB), B_6(kB)</math> are in Morita equivalence classes not listed in the table.