Difference between revisions of "D8"
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== Blocks with defect group <math>D_8</math> == | == Blocks with defect group <math>D_8</math> == | ||
| − | These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er87] ]]). The classification with respect to <math>\mathcal{O}</math> is still unknown. | + | These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References#E|[Er87] ]]). The classification with respect to <math>\mathcal{O}</math> is still unknown. |
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! scope="col"| Class | ! scope="col"| Class | ||
! scope="col"| Representative | ! scope="col"| Representative | ||
| + | ! scope="col"| # lifts / <math>\mathcal{O}</math> | ||
! scope="col"| <math>k(B)</math> | ! scope="col"| <math>k(B)</math> | ||
! scope="col"| <math>l(B)</math> | ! scope="col"| <math>l(B)</math> | ||
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|- | |- | ||
| − | |<math>M(8,3,1 | + | |[[M(8,3,1)]] || <math>kD_8</math> || 1 ||5 ||1 ||<math>1</math> || || || ||1 || |
| + | |- | ||
| + | |[[M(8,3,2)]] || <math>B_0(kPGL_2(5))</math> || ? ||5 ||2 ||<math>1</math> || || || ||1 || <math>D(2 {\cal A})</math> | ||
| + | |- | ||
| + | |[[M(8,3,3)]] || <math>kS_4</math> || ? ||5 ||2 ||<math>1</math> || || || ||1 || <math>D(2 {\cal B})</math> | ||
|- | |- | ||
| − | | | + | |[[M(8,3,4)]] || <math>B_0(kPSL_2(9))</math> || ? ||5 ||3 ||<math>1</math> || || || ||1 || <math>D(3 {\cal A})_1</math> |
|- | |- | ||
| − | | | + | |[[M(8,3,5)]] || <math>B_0(kA_7)</math> || ? ||5 ||3 ||<math>1</math> || || || ||1 || <math>D(3 {\cal B})_1</math> |
|- | |- | ||
| − | | | + | |[[M(8,3,6)]] || <math>B_0(kPSL_2(7))</math> || ? ||5 ||3 ||<math>1</math> || || || ||1 || <math>D(3 {\cal K})</math> |
|} | |} | ||
| − | + | [[M(8,3,2)]] and [[M(8,3,3)]] are derived equivalent over <math>k</math> by [[References#H|[Ho97] ]]. | |
| + | |||
| + | [[M(8,3,4)]], [[M(8,3,5)]] and [[M(8,3,6)]] are derived equivalent over <math>k</math> by [[References#L|[Li94b] ]]. | ||
Latest revision as of 16:54, 18 December 2018
Blocks with defect group [math]D_8[/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.
| 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(8,3,1) | [math]kD_8[/math] | 1 | 5 | 1 | [math]1[/math] | 1 | ||||
| M(8,3,2) | [math]B_0(kPGL_2(5))[/math] | ? | 5 | 2 | [math]1[/math] | 1 | [math]D(2 {\cal A})[/math] | |||
| M(8,3,3) | [math]kS_4[/math] | ? | 5 | 2 | [math]1[/math] | 1 | [math]D(2 {\cal B})[/math] | |||
| M(8,3,4) | [math]B_0(kPSL_2(9))[/math] | ? | 5 | 3 | [math]1[/math] | 1 | [math]D(3 {\cal A})_1[/math] | |||
| M(8,3,5) | [math]B_0(kA_7)[/math] | ? | 5 | 3 | [math]1[/math] | 1 | [math]D(3 {\cal B})_1[/math] | |||
| M(8,3,6) | [math]B_0(kPSL_2(7))[/math] | ? | 5 | 3 | [math]1[/math] | 1 | [math]D(3 {\cal K})[/math] |
M(8,3,2) and M(8,3,3) are derived equivalent over [math]k[/math] by [Ho97] .
M(8,3,4), M(8,3,5) and M(8,3,6) are derived equivalent over [math]k[/math] by [Li94b] .
