Difference between revisions of "Q16"
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== Blocks with defect group <math>Q_{16}</math> == | == Blocks with defect group <math>Q_{16}</math> == | ||
| − | These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[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. | + | These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er88a], [Er88b]]]) with some exceptions.<ref>Note that as in [[References|[Ho97]]] the class <math>Q(2 {\cal B})_2</math> cannot be realised by a block.</ref> 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 [[References|[Ei16]]], but otherwise the classification with respect to <math>\mathcal{O}</math> is still unknown. | For blocks with three simple modules the <math>k</math>-Morita equivalence classes lift to unique <math>\mathcal{O}</math>-classes by [[References|[Ei16]]], but otherwise the classification with respect to <math>\mathcal{O}</math> is still unknown. | ||
Revision as of 16:02, 2 April 2025
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.[1] 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][2] | ? | 8 | 2 | [math]1[/math] | 1 | [math]Q(2 {\cal A})[/math] | |||
| M(16,9,3) | [math]B_0(k \tilde{S}_4)[/math][3] | ? | 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][4].
