# Q16

## 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.

CLASSIFICATION INCOMPLETE

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]}.