Difference between revisions of "D8"

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(Created page with "== 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 References|[...")
 
(Added classes with 2 simple modules)
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|<math>M(8,3,1)</math> || <math>k(D_8)</math> ||5 ||1 ||<math>1</math> || || || ||1 ||
 
|<math>M(8,3,1)</math> || <math>k(D_8)</math> ||5 ||1 ||<math>1</math> || || || ||1 ||
 
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|<math>M(8,3,2)</math> || <math>B_0(kPSL_2(9))</math> ||5 ||3 ||<math>1</math> || || || ||1 || <math>D(3 {\cal A})_1</math>
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|<math>M(8,3,2)</math> || <math>B_0(kPGL_2(5))</math> ||5 ||2 ||<math>1</math> || || || ||1 || <math>D(2 {\cal A})</math>
 
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|<math>M(8,3,3)</math> || <math>B_0(kA_7)</math> ||5 ||3 ||<math>1</math> || || || ||1 || <math>D(3 {\cal B})_1</math>
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|<math>M(8,3,3)</math> || <math>kS_4</math> ||5 ||2 ||<math>1</math> || || || ||1 || <math>D(2 {\cal B})</math>
 
|-
 
|-
|<math>M(8,3,4)</math> || <math>B_0(kPSL_2(7))</math> ||5 ||3 ||<math>1</math> || || || ||1 || <math>D(3 {\cal K})</math>
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|<math>M(8,3,4)</math> || <math>B_0(kPSL_2(9))</math> ||5 ||3 ||<math>1</math> || || || ||1 || <math>D(3 {\cal A})_1</math>
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|<math>M(8,3,5)</math> || <math>B_0(kA_7)</math> ||5 ||3 ||<math>1</math> || || || ||1 || <math>D(3 {\cal B})_1</math>
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|<math>M(8,3,6)</math> || <math>B_0(kPSL_2(7))</math> ||5 ||3 ||<math>1</math> || || || ||1 || <math>D(3 {\cal K})</math>
 
|}
 
|}
  
<math>M(8,3,2)</math>, <math>M(8,3,3)</math> and <math>M(8,3,3)</math> are derived equivalent over <math>k</math> by [[References|[Li94b] ]].
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<math>M(8,3,2)</math>, <math>M(8,3,3)</math> and <math>M(8,3,4)</math> are derived equivalent over <math>k</math> by [[References|[Ho97] ]].
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<math>M(8,3,4)</math>, <math>M(8,3,5)</math> and <math>M(8,3,6)</math> are derived equivalent over <math>k</math> by [[References|[Li94b] ]].

Revision as of 13:48, 28 August 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 [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
[math]M(8,3,1)[/math] [math]k(D_8)[/math] 5 1 [math]1[/math] 1
[math]M(8,3,2)[/math] [math]B_0(kPGL_2(5))[/math] 5 2 [math]1[/math] 1 [math]D(2 {\cal A})[/math]
[math]M(8,3,3)[/math] [math]kS_4[/math] 5 2 [math]1[/math] 1 [math]D(2 {\cal B})[/math]
[math]M(8,3,4)[/math] [math]B_0(kPSL_2(9))[/math] 5 3 [math]1[/math] 1 [math]D(3 {\cal A})_1[/math]
[math]M(8,3,5)[/math] [math]B_0(kA_7)[/math] 5 3 [math]1[/math] 1 [math]D(3 {\cal B})_1[/math]
[math]M(8,3,6)[/math] [math]B_0(kPSL_2(7))[/math] 5 3 [math]1[/math] 1 [math]D(3 {\cal K})[/math]

[math]M(8,3,2)[/math], [math]M(8,3,3)[/math] and [math]M(8,3,4)[/math] are derived equivalent over [math]k[/math] by [Ho97] .

[math]M(8,3,4)[/math], [math]M(8,3,5)[/math] and [math]M(8,3,6)[/math] are derived equivalent over [math]k[/math] by [Li94b] .