Difference between revisions of "C3xC3"
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| − | 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. | + | 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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|[[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 || 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(9,2,16)]] || <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> || ||9 ||7 ||<math>SD_{16}</math> || || ||1 ||1 || Extension by graph automorphism | |[[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)]] || <math> | + | |[[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_{ | + | |[[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 || |
| + | |- | ||
| + | |[[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_3(kCo_1)</math> || || 9 || 5 || <math>D_8</math> || || || 1 || 1 || | ||
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| || Block of <math>kFi_{24}'.2</math> covering <math>B_2(kFi_{24}')</math> || || 9 || 5 || <math>D_8</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> | + | | || <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_6(k(2.M_{22}))</math> || || 6 || 5 || || || || 1 || 1 || | ||
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Some open problems: | 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_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. | *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]
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.
