D8

Blocks with defect group [math]D_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.

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(8,3,1)	[math]k(D_8)[/math]	1	5	1	[math]1[/math]				1
M(8,3,2)	[math]B_0(kPGL_2(5))[/math]	?	5	2	[math]1[/math]				1	[math]D(2 {\cal A})[/math]
M(8,3,3)	[math]kS_4[/math]	?	5	2	[math]1[/math]				1	[math]D(2 {\cal B})[/math]
M(8,3,4)	[math]B_0(kPSL_2(9))[/math]	?	5	3	[math]1[/math]				1	[math]D(3 {\cal A})_1[/math]
M(8,3,5)	[math]B_0(kA_7)[/math]	?	5	3	[math]1[/math]				1	[math]D(3 {\cal B})_1[/math]
M(8,3,6)	[math]B_0(kPSL_2(7))[/math]	?	5	3	[math]1[/math]				1	[math]D(3 {\cal K})[/math]

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

M(8,3,4), M(8,3,5) and M(8,3,6) are derived equivalent over [math]k[/math] by [Li94b] .