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 Morita equivalence classes lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References|[HKL07], [Ei16]]] 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#E|[Er87] ]]). The Morita equivalence classes lift to unique Morita equivalence classes over <math>\mathcal{O}</math> by [[References#H|[HKL07]]], [[References#E|[Ei16]]] and the theory of [[nilpotent blocks]]. |
{| class="wikitable" | {| class="wikitable" | ||
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| − | |[[M(8,4,1)]] || <math>kQ_8</math> || 1 ||5 ||1 ||<math>1</math> || || ||1 ||1 || | + | |[[M(8,4,1)]] || <math>kQ_8</math> || 1 ||5 ||1 ||<math>1</math> || <math>S_4</math> || ||1 ||1 || |
|- | |- | ||
| − | |[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> || 1 ||7 ||3 ||<math>C_3</math> || || || ||1 || <math>Q(3 {\cal A})_2</math> | + | |[[M(8,4,2)]] || <math>B_0(kSL_2(5))</math> || 1 ||7 ||3 ||<math>C_3</math> || <math>C_2</math> || || ||1 || <math>Q(3 {\cal A})_2</math> |
|- | |- | ||
| − | |[[M(8,4,3)]] || <math>kSL_2(3)</math> || 1 ||7 ||3 ||<math>C_3</math> || || ||1 ||1 || <math>Q(3 {\cal K})</math> | + | |[[M(8,4,3)]] || <math>kSL_2(3)</math> || 1 ||7 ||3 ||<math>C_3</math> || <math>S_3</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#H|[Ho97] ]]. |
Latest revision as of 08:35, 24 May 2022
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 Morita equivalence classes lift to unique Morita equivalence classes over [math]\mathcal{O}[/math] by [HKL07], [Ei16] 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] | [math]S_4[/math] | 1 | 1 | ||
| M(8,4,2) | [math]B_0(kSL_2(5))[/math] | 1 | 7 | 3 | [math]C_3[/math] | [math]C_2[/math] | 1 | [math]Q(3 {\cal A})_2[/math] | ||
| M(8,4,3) | [math]kSL_2(3)[/math] | 1 | 7 | 3 | [math]C_3[/math] | [math]S_3[/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] .
