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

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

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