Difference between revisions of "C5"
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== Blocks with defect group <math>C_5</math> == | == Blocks with defect group <math>C_5</math> == | ||
| − | These are blocks with | + | 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. | There are six <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the possible Brauer trees. | ||
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|[[M(5,1,1)]] || <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]] | |[[M(5,1,1)]] || <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]] | ||
|- | |- | ||
| − | |[[M(5,1,2)]] || <math>kD_{10}</math> ||4 ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(5,1,2)tree.png|45px]] | + | |[[M(5,1,2)]] || <math>kD_{10}</math> ||4 ||2 ||<math>C_2</math> || <math>C_2 \times C_2</math> || ||1 ||1 || [[Image:M(5,1,2)tree.png|45px]] |
|- | |- | ||
| − | |[[M(5,1,3)]] || <math>B_0(kA_5)</math> ||4 ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(5,1,3)tree.png|45px]] | + | |[[M(5,1,3)]] || <math>B_0(kA_5)</math> ||4 ||2 ||<math>C_2</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(5,1,3)tree.png|45px]] |
|- | |- | ||
|[[M(5,1,4)]] || <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]] | |[[M(5,1,4)]] || <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]] | ||
|- | |- | ||
| − | |[[M(5,1,5)]] || <math>B_0(kA_7)</math> ||5 ||4 ||<math>C_4</math> || || ||1 ||1 || [[Image:M(5,1,5)tree.png|45px]] | + | |[[M(5,1,5)]] || <math>B_0(kA_7)</math> ||5 ||4 ||<math>C_4</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(5,1,5)tree.png|45px]] |
|- | |- | ||
| − | |[[M(5,1,6)]] || | + | |[[M(5,1,6)]] || <math>B_{15}(k(6.A_7))</math> ||5 ||4 ||<math>C_4</math> || <math>1</math> || ||1 ||1 || [[Image:M(5,1,6)tree.png|45px]] |
|} | |} | ||
Latest revision as of 10:28, 22 November 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 |
|---|---|---|---|---|---|---|---|---|---|
| M(5,1,1) | [math]kC_5[/math] | 5 | 1 | [math]1[/math] | [math]C_5:C_4[/math] | 1 | 1 | tree.png){kind=small}
| |
| M(5,1,2) | [math]kD_{10}[/math] | 4 | 2 | [math]C_2[/math] | [math]C_2 \times C_2[/math] | 1 | 1 | tree.png){kind=small}
| |
| M(5,1,3) | [math]B_0(kA_5)[/math] | 4 | 2 | [math]C_2[/math] | [math]C_2[/math] | 1 | 1 | tree.png){kind=small}
| |
| M(5,1,4) | [math]k(C_5:C_4)[/math] | 5 | 4 | [math]C_4[/math] | [math]C_4[/math] | 1 | 1 | tree.png){kind=small}
| |
| M(5,1,5) | [math]B_0(kA_7)[/math] | 5 | 4 | [math]C_4[/math] | [math]C_2[/math] | 1 | 1 | tree.png){kind=small}
| |
| M(5,1,6) | [math]B_{15}(k(6.A_7))[/math] | 5 | 4 | [math]C_4[/math] | [math]1[/math] | 1 | 1 | tree.png){kind=small}
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