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> ==
  
[[Image:under-construction.png|50px|left]]
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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]].
 
 
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]].
 
  
 
{| class="wikitable"
 
{| class="wikitable"
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|-
 
|-
|[[M(8,4,1)]] || <math>kQ_8</math> || 1 ||5 ||1 ||<math>1</math> || || ||1 ||1 ||
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|[[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> || ? ||5 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
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|[[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 ||5 ||3 ||<math>1</math> || || ||1 ||1 || <math>Q(3 {\cal K})</math>
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|[[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] ]].
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[[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] .