# D32

## Blocks with defect group [math]D_{32}[/math]

These are examples of tame blocks and were first classified over [math]k[/math] by Erdmann (see [Er87] ). The classification with respect to [math]\mathcal{O}[/math] is still unknown. Note that the class [math]D(3 {\cal B})_1[/math] is only realised for defect groups [math]D_8[/math] (see [Er87, Proposition 7.5.1]).

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(32,18,1) | [math]k(D_{32})[/math] | 1 | 11 | 1 | [math]1[/math] | 1 | ||||

M(32,18,2) | [math]B_0(kPGL_2(17))[/math] | ? | 11 | 2 | [math]1[/math] | 1 | [math]D(2 {\cal A})[/math] | |||

M(32,18,3) | [math]B_0(kPGL_2(47))[/math] | ? | 11 | 2 | [math]1[/math] | 1 | [math]D(2 {\cal B})[/math] | |||

M(32,18,4) | [math]B_0(kPSL_2(97))[/math] | ? | 11 | 3 | [math]1[/math] | 1 | [math]D(3 {\cal A})_1[/math] | |||

M(32,18,5) | [math]B_0(kPSL_2(31))[/math] | ? | 11 | 3 | [math]1[/math] | 1 | [math]D(3 {\cal K})[/math] |

M(32,18,2) and M(32,18,3) are derived equivalent over [math]k[/math] by [Ho97] .

M(32,18,4) and M(32,18,5) are derived equivalent over [math]k[/math] by [Li94b] .