D8

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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(kPSL_2(9))[/math] 5 3 [math]1[/math] 1 [math]D(3 {\cal A})_1[/math]
[math]M(8,3,3)[/math] [math]B_0(kA_7)[/math] 5 3 [math]1[/math] 1 [math]D(3 {\cal B})_1[/math]
[math]M(8,3,4)[/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(8,3,3)[/math] are derived equivalent over [math]k[/math] by [Li94b] .