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Blocks with defect group [math]Q_{16}[/math]

These are examples of tame blocks and were first classified over [math]k[/math] by Erdmann (see [Er88a], [Er88b]) with some exceptions. It is not known which algebras in the infinite families [math]Q(2 {\cal A})[/math] and [math]Q(2 {\cal B})_1[/math] are realised by blocks, and as such Donovan's conjecture is still open for [math]Q_{16}[/math] for blocks with two simple modules. Until this is resolved the labelling is provisional.

For blocks with three simple modules the [math]k[/math]-Morita equivalence classes lift to unique [math]\mathcal{O}[/math]-classes by [Ei16], but otherwise 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(16,9,1) [math]kSD_{16}[/math] 1 7 1 [math]1[/math] 1
M(16,9,2) [math]B_0(k \tilde{S}_5)[/math][1]  ? 8 2 [math]1[/math] 1 [math]Q(2 {\cal A})[/math]
M(16,9,3) [math]B_0(k \tilde{S}_4)[/math][2]  ? 8 2 [math]1[/math] 1 [math]Q(2 {\cal B})_1[/math]
M(16,9,4) [math]B_0(kSL_2(9))[/math] 1 9 3 [math]1[/math] 1 [math]Q(3 {\cal A})_2[/math]
M(16,9,5) [math]B_0(k(2.A_7))[/math] 1 10 3 [math]1[/math] 1 [math]Q(3 {\cal B})[/math]
M(16,9,6) [math]B_0(kSL_2(7))[/math] 1 9 3 [math]1[/math] 1 [math]Q(3 {\cal K})[/math]

M(16,9,2) and M(16,9,3) are derived equivalent over [math]k[/math] by [Ho97], in which it is further proved that all blocks with defect group [math]Q_{16}[/math] and two simple modules are derived equivalent (irrespective of the unknown cases in the classification).

M(16,9,4), M(16,9,5) and M(16,9,6) are derived equivalent over [math]\mathcal{O}[/math] by [Ei16][3].


  1. This is the double cover SmallGroup(240,89)
  2. This is the double cover SmallGroup(48,28)
  3. This result was obtained over [math]k[/math] in [Ho97]