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.

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]			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)	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.