Q8

Blocks with defect group [math]Q_8[/math]

These are examples of tame blocks and were first classified over [math]k[/math] by Erdmann (see [Er87] ). The classification with respect to [math]\mathcal{O}[/math] is still unknown in general, except for the cases M(8,4,1) and M(8,4,3), which both lift to unique Morita equivalence classes over [math]\mathcal{O}[/math] by [HKL07] and the theory of nilpotent blocks.

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(8,4,1)	[math]kQ_8[/math]	1	5	1	[math]1[/math]			1	1
M(8,4,2)	[math]B_0(kSL_2(5))[/math]	?	5	3	[math]1[/math]				1	[math]Q(3 {\cal A})_2[/math]
M(8,4,3)	[math]kSL_2(3)[/math]	1	5	3	[math]1[/math]			1	1	[math]Q(3 {\cal K})[/math]

M(8,4,2) and M(8,4,3) are derived equivalent over [math]k[/math] by [Ho97] .