Difference between revisions of "Q8"
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== Blocks with defect group <math>Q_8</math> == | == Blocks with defect group <math>Q_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 in general, except for the cases [[M(8,4,1)]] and [[M(8,4,3)]], which both lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References|[HKL07] ]] and the theory of [[nilpotent blocks]]. | 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 in general, except for the cases [[M(8,4,1)]] and [[M(8,4,3)]], which both lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References|[HKL07] ]] and the theory of [[nilpotent blocks]]. | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 8: | Line 9: | ||
! 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> | ||
| Line 18: | Line 20: | ||
|- | |- | ||
| − | |[[M(8,4,1)]] || <math>kQ_8</math> ||5 ||1 ||<math>1</math> || || ||1 ||1 || | + | |[[M(8,4,1)]] || <math>kQ_8</math> || 1 ||5 ||1 ||<math>1</math> || || ||1 ||1 || |
|- | |- | ||
| − | |[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> ||5 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math> | + | |[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> || ? ||5 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math> |
|- | |- | ||
| − | |[[M(8,4,3)]] || <math>kSL_2(3)</math> ||5 ||3 ||<math>1</math> || || ||1 ||1 || <math>Q(3 {\cal K})</math> | + | |[[M(8,4,3)]] || <math>kSL_2(3)</math> || 1 ||5 ||3 ||<math>1</math> || || ||1 ||1 || <math>Q(3 {\cal K})</math> |
|} | |} | ||
[[M(8,4,2)]] and [[M(8,4,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]]. | [[M(8,4,2)]] and [[M(8,4,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]]. | ||
Revision as of 08:15, 15 September 2018
Blocks with defect group [math]Q_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 in general, except for the cases M(8,4,1) and M(8,4,3), which both lift to unique Morita equivalence classes over [math]\mathcal{O}[/math] by [HKL07] and the theory of nilpotent blocks.
| 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(8,4,1) | [math]kQ_8[/math] | 1 | 5 | 1 | [math]1[/math] | 1 | 1 | |||
| M(8,4,2) | [math]B_0(kSL_2(5))[/math] | ? | 5 | 3 | [math]1[/math] | 1 | [math]Q(3 {\cal A})_2[/math] | |||
| M(8,4,3) | [math]kSL_2(3)[/math] | 1 | 5 | 3 | [math]1[/math] | 1 | 1 | [math]Q(3 {\cal K})[/math] |
M(8,4,2) and M(8,4,3) are derived equivalent over [math]k[/math] by [Ho97] .
