Q16

Blocks with defect group [math]Q_{16}[/math]

These are examples of tame blocks and were first classified over [math]k[/math] by Erdmann (see [Er88a], [Er88b]) with some exceptions. It is not known which algebras in the infinite families [math]Q(2 {\cal A})[/math] and [math]Q(2 {\cal B})_1[/math] are realised by blocks, and as such Donovan's conjecture is still open for [math]Q_{16}[/math] for blocks with two simple modules. Until this is resolved the labelling is provisional.

For blocks with three simple modules the [math]k[/math]-Morita equivalence classes lift to unique [math]\mathcal{O}[/math]-classes by [Ei16], but otherwise the classification with respect to [math]\mathcal{O}[/math] is still unknown.

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(16,9,1)	[math]kSD_{16}[/math]	1	7	1	[math]1[/math]				1
M(16,9,2)	[math]B_0(k \tilde{S}_5)[/math]	?	8	2	[math]1[/math]				1	[math]Q(2 {\cal A})[/math]
M(16,9,3)	[math]B_0(k \tilde{S}_4)[/math]	?	8	2	[math]1[/math]				1	[math]Q(2 {\cal B})_1[/math]
M(16,9,4)	[math]B_0(kSL_2(9))[/math]	1	9	3	[math]1[/math]				1	[math]Q(3 {\cal A})_2[/math]
M(16,9,5)	[math]B_0(k(2.A_7))[/math]	1	10	3	[math]1[/math]				1	[math]Q(3 {\cal B})[/math]
M(16,9,6)	[math]B_0(kSL_2(7))[/math]	1	9	3	[math]1[/math]				1	[math]Q(3 {\cal K})[/math]

M(16,9,2) and M(16,9,3) are derived equivalent over [math]k[/math] by [Ho97], in which it is further proved that all blocks with defect group [math]Q_{16}[/math] and two simple modules are derived equivalent (irrespective of the unknown cases in the classification).

M(16,9,4), M(16,9,5) and M(16,9,6) are derived equivalent over [math]\mathcal{O}[/math] by [Ei16].