Difference between revisions of "C2xC2xC2"
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== Blocks with defect group <math>C_2 \times C_2 \times C_2</math> == | == Blocks with defect group <math>C_2 \times C_2 \times C_2</math> == | ||
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These were classified in [[References|[Ea16]]] using the [[Glossary#CFSG|CFSG]]. Each of the eight <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. | These were classified in [[References|[Ea16]]] using the [[Glossary#CFSG|CFSG]]. Each of the eight <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. | ||
Revision as of 10:41, 5 October 2018
Blocks with defect group [math]C_2 \times C_2 \times C_2[/math]
These were classified in [Ea16] using the CFSG. Each of the eight [math]k[/math]-Morita equivalence classes lifts to an unique class over [math]\mathcal{O}[/math].
| 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(8,5,1) | [math]k(C_2 \times C_2 \times C_2)[/math] | 1 | 8 | 1 | [math]1[/math] | 1 | 1 | |||
| M(8,5,2) | [math]B_0(k(A_5 \times C_2))[/math] | 1 | 8 | 3 | [math]C_3[/math] | 1 | 1 | |||
| M(8,5,3) | [math]k(A_4 \times C_2)[/math] | 1 | 8 | 3 | [math]C_3[/math] | 1 | 1 | |||
| M(8,5,4) | [math]k((C_2 \times C_2 \times C_2):C_7)[/math] | 1 | 8 | 7 | [math]C_7[/math] | 1 | 1 | |||
| M(8,5,5) | [math]B_0(kSL_2(8))[/math] | 1 | 8 | 7 | [math]C_7[/math] | 1 | 1 | |||
| M(8,5,6) | [math]k((C_2 \times C_2 \times C_2):(C_7:C_3))[/math] | 1 | 8 | 5 | [math]C_7:C_3[/math] | 1 | 1 | |||
| M(8,5,7) | [math]B_0(kJ_1)[/math] | 1 | 8 | 5 | [math]C_7:C_3[/math] | 1 | 1 | |||
| M(8,5,8) | [math]B_0(k{\rm Aut}(SL_2(8)))[/math] | 1 | 8 | 5 | [math]C_7:C_3[/math] | 1 | 1 |
M(8,5,2) and M(8,5,3) are derived equivalent over [math]\mathcal{O}[/math].
M(8,5,4) and M(8,5,5) are derived equivalent over [math]\mathcal{O}[/math].
M(8,5,6), M(8,5,7) and M(8,5,8) are derived equivalent over [math]\mathcal{O}[/math].
