Difference between revisions of "C5"
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| − | |[[M(5,1,1)|<math>M(5,1,1)</math>]] || <math>kC_5</math> ||5 ||1 ||<math>1</math> ||<math>C_5:C_4</math> || ||1 ||1 || | + | |[[M(5,1,1)|<math>M(5,1,1)</math>]] || <math>kC_5</math> ||5 ||1 ||<math>1</math> ||<math>C_5:C_4</math> || ||1 ||1 || [[Image:M(5,1,1)tree.png|45px]] |
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| − | |[[M(5,1,2)|<math>M(5,1,2)</math>]] || <math>kD_{10}</math> ||4 ||2 ||<math>C_2</math> || || ||1 ||1 || | + | |[[M(5,1,2)|<math>M(5,1,2)</math>]] || <math>kD_{10}</math> ||4 ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(5,1,2)tree.png|45px]] |
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| − | |[[M(5,1,3)|<math>M(5,1,3)</math>]] || <math>B_0(kA_5)</math> ||4 ||2 ||<math>C_2</math> || || ||1 ||1 || | + | |[[M(5,1,3)|<math>M(5,1,3)</math>]] || <math>B_0(kA_5)</math> ||4 ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(5,1,3)tree.png|45px]] |
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| − | |[[M(5,1,4)|<math>M(5,1,4)</math>]] || <math>k(C_5:C_4)</math> ||5 ||4 ||<math>C_4</math> ||<math>C_4</math> || ||1 ||1 || | + | |[[M(5,1,4)|<math>M(5,1,4)</math>]] || <math>k(C_5:C_4)</math> ||5 ||4 ||<math>C_4</math> ||<math>C_4</math> || ||1 ||1 || [[Image:M(5,1,4)tree.png|45px]] |
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| − | |[[M(5,1,5)|<math>M(5,1,5)</math>]] || <math>B_0(kA_7)</math> ||5 ||4 ||<math>C_4</math> || || ||1 ||1 || | + | |[[M(5,1,5)|<math>M(5,1,5)</math>]] || <math>B_0(kA_7)</math> ||5 ||4 ||<math>C_4</math> || || ||1 ||1 || [[Image:M(5,1,5)tree.png|45px]] |
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| − | |[[M(5,1,6)|<math>M(5,1,6)</math>]] || faithful block of <math>k(2.A_7)</math> ||5 ||4 ||<math>C_4</math> |||| ||1 ||1 || | + | |[[M(5,1,6)|<math>M(5,1,6)</math>]] || faithful block of <math>k(2.A_7)</math> ||5 ||4 ||<math>C_4</math> |||| ||1 ||1 || [[Image:M(5,1,6)tree.png|45px]] |
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Revision as of 20:51, 3 September 2018
Blocks with defect group [math]C_5[/math]
These are blocks with cyclic defect groups and so they are described by Brauer trees.
There are six [math]\mathcal{O}[/math]-Morita equivalence classes, accounting for all the possible Brauer trees.
| 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(5,1,1)[/math] | [math]kC_5[/math] | 5 | 1 | [math]1[/math] | [math]C_5:C_4[/math] | 1 | 1 | tree.png){kind=small}
| |
| [math]M(5,1,2)[/math] | [math]kD_{10}[/math] | 4 | 2 | [math]C_2[/math] | 1 | 1 | tree.png){kind=small}
| ||
| [math]M(5,1,3)[/math] | [math]B_0(kA_5)[/math] | 4 | 2 | [math]C_2[/math] | 1 | 1 | tree.png){kind=small}
| ||
| [math]M(5,1,4)[/math] | [math]k(C_5:C_4)[/math] | 5 | 4 | [math]C_4[/math] | [math]C_4[/math] | 1 | 1 | tree.png){kind=small}
| |
| [math]M(5,1,5)[/math] | [math]B_0(kA_7)[/math] | 5 | 4 | [math]C_4[/math] | 1 | 1 | tree.png){kind=small}
| ||
| [math]M(5,1,6)[/math] | faithful block of [math]k(2.A_7)[/math] | 5 | 4 | [math]C_4[/math] | 1 | 1 | tree.png){kind=small}
|
