Difference between revisions of "C3xC3"

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[[Image:under-construction.png|50px|left]]
 
[[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. 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>'''
 
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''
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! scope="col"| Class
 
! scope="col"| Class
 
! scope="col"| Representative
 
! 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>k(B)</math>
 
! scope="col"| <math>l(B)</math>
 
! scope="col"| <math>l(B)</math>
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(9,2,1)]] || <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 ||
+
|[[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(9,2,16)]] ||  <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 ||
 
|[[M(9,2,18)]] ||  <math>B_0(kM_{11})</math> ||  ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 ||
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|[[M(9,2,19)]] ||  <math>B_0(kHS)</math> ||  ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 ||
 
|[[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> (adjoining graph auto) ||  ||9 ||7 ||<math>SD_{16}</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)
 
|[[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)]] || 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_{10}(k(4.M_{22}))</math> ||  ||6 ||5 ||<math>Q_8</math> || || ||1 ||1 ||
 +
|-
 +
|[[M(9,2,26)]] || <math>B_{7}(k(2.HS))</math> ||  ||9 ||5 ||<math>D_8</math> || || ||1 ||1 ||
 +
|-
 +
|[[M(9,2,27)]] || <math>B_{2}(k(HS))</math> ||  ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 ||
 +
|-
 +
| || <math>B_3(kCo_1)</math> || || 9 || 5 || <math>D_8</math> || || || 1 || 1 ||
 +
|-
 +
| || <math>B_6(kJ_4)</math> || || 9 || 5 || <math>D_8</math> || || || 1 || 1 ||
 +
|-
 +
| || <math>B_2(kFi_{24}')</math> || || 6 || 4 || <math>C_4</math> || || || 1 || 1 ||
 +
|-
 +
| || Block of <math>kFi_{24}'.2</math> covering <math>B_2(kFi_{24}')</math> || || 9 || 5 || <math>D_8</math> || || || 1 || 1 ||
 +
|-
 +
| || <math>B_3(k(HS.2))</math> || || 9 || 5 || <math>D_8</math> || || || 1 || 1 ||
 +
|-
 +
| || <math>B_{12}(k(2.HS.2))</math> || || 9 || 7 || <math>SD_{16}</math> || || || 1 || 1 ||
 +
|-
 +
| || <math>B_6(k(2.M_{22}))</math> || || 6 || 5 || || || || 1 || 1 ||
 +
|-
 +
| || <math>B_9(k(2.M_{22}.2))</math> || || 9 || 7 || <math>SD_{16}</math> || || || 1 || 1 ||
 +
|-
 +
| || <math>B_6(kB)</math> || || 9 || 5 || <math>D_8</math> || || || 1 || 1 ||
 +
|-
 +
| || <math>B_2(kB)</math> || || 9 || 7 || <math>D_8</math> || || || 1 || 1 ||
 +
|-
 +
| || <math>B_3(kB)</math> || || 9 || 7 || <math>D_8</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_0(kA_8)</math> is Morita equivalent to <math>B_3(k(HS.2))</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.

Latest revision as of 09:45, 24 May 2022

Blocks with defect group [math]C_3 \times C_3[/math]

Under-construction.png

Source algebra equivalence classes of principal blocks with this defect group have been classified by Koshitani in [Ko03] using the CFSG. This accounts for classes M(9,2,1) to M(9,2,22). Note that it does not follow from the results in [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 [Mar].

The numerical invariants of arbitrary blocks with defect group [math]C_3 \times C_3[/math] were calculated for all inertial quotients except [math]C_8[/math] and [math]Q_8[/math] by Kiyota in [Ki84], and these cases remain an open problem. Kiyota's calculations do not involve the CFSG.

CLASSIFICATION INCOMPLETE
Class Representative # lifts / [math]\mathcal{O}[/math] [math]k(B)[/math] [math]l(B)[/math] Inertial quotients [math]{\rm Pic}_\mathcal{O}(B)[/math] [math]{\rm Pic}_k(B)[/math] [math]{\rm mf_\mathcal{O}(B)}[/math] [math]{\rm mf_k(B)}[/math] Notes
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] [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) [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] [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] [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] [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] [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] [math]S_4[/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] [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] [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] [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] [math]S_4[/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] [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
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] [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] [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)
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_{10}(k(4.M_{22}))[/math] 6 5 [math]Q_8[/math] 1 1
M(9,2,26) [math]B_{7}(k(2.HS))[/math] 9 5 [math]D_8[/math] 1 1
M(9,2,27) [math]B_{2}(k(HS))[/math] 9 7 [math]SD_{16}[/math] 1 1
[math]B_3(kCo_1)[/math] 9 5 [math]D_8[/math] 1 1
[math]B_6(kJ_4)[/math] 9 5 [math]D_8[/math] 1 1
[math]B_2(kFi_{24}')[/math] 6 4 [math]C_4[/math] 1 1
Block of [math]kFi_{24}'.2[/math] covering [math]B_2(kFi_{24}')[/math] 9 5 [math]D_8[/math] 1 1
[math]B_3(k(HS.2))[/math] 9 5 [math]D_8[/math] 1 1
[math]B_{12}(k(2.HS.2))[/math] 9 7 [math]SD_{16}[/math] 1 1
[math]B_6(k(2.M_{22}))[/math] 6 5 1 1
[math]B_9(k(2.M_{22}.2))[/math] 9 7 [math]SD_{16}[/math] 1 1
[math]B_6(kB)[/math] 9 5 [math]D_8[/math] 1 1
[math]B_2(kB)[/math] 9 7 [math]D_8[/math] 1 1
[math]B_3(kB)[/math] 9 7 [math]D_8[/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_0(kA_8)[/math] is Morita equivalent to [math]B_3(k(HS.2))[/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.