C3xC3

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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.

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. These 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] 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(9,2,4) [math]k(S_3 \times S_3)[/math] 9 4 [math]C_2 \times C_2[/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(9,2,6) [math]B_0(kA_6)[/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] 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(9,2,9) [math]B_0(kPGL_2(9))[/math] 9 8 [math]C_8[/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(9,2,11) [math]B_0(kA_8)[/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] 1 1
M(9,2,13) [math]B_0(S_7)[/math] 9 5 [math]D_8[/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(9,2,15) [math]B_0(kM_{22})[/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(9,2,18) [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] 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(9,2,23) 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(9,2,25) [math]B_{7}(k(2.HS))[/math] 9 5 [math]D_8[/math] 1 1
M(9,2,26) [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_2(k(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_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.