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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 Morita equivalence classes lift to unique Morita equivalence classes over [math]\mathcal{O}[/math] by [HKL07], [Ei16] 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] [math]S_4[/math] 1 1
M(8,4,2) [math]B_0(kSL_2(5))[/math] 1 7 3 [math]C_3[/math] [math]C_2[/math] 1 [math]Q(3 {\cal A})_2[/math]
M(8,4,3) [math]kSL_2(3)[/math] 1 7 3 [math]C_3[/math] [math]S_3[/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] .