C3xC3
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.
