Difference between revisions of "C3"
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== Blocks with defect group <math>C_3</math> == | == Blocks with defect group <math>C_3</math> == | ||
| − | These are blocks with | + | These are [[blocks with cyclic defect groups]] and so they are described by [[Brauer trees]]. |
There are two <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the possible Brauer trees. | There are two <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the possible Brauer trees. | ||
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! scope="col"| Class | ! scope="col"| Class | ||
! scope="col"| Representative | ! scope="col"| Representative | ||
| + | ! scope="col"| # lifts / <math>\mathcal{O}</math> | ||
! scope="col"| <math>k(B)</math> | ! scope="col"| <math>k(B)</math> | ||
! scope="col"| <math>l(B)</math> | ! scope="col"| <math>l(B)</math> | ||
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|- | |- | ||
| − | | | + | |[[M(3,1,1)]] || <math>kC_3</math> || 1 ||3 ||1 ||<math>1</math> ||<math>S_3</math> || <math>k:k^*</math> ||1 ||1 || [[Image:M(3,1,1)tree.png|45px]] |
|- | |- | ||
| − | | | + | |[[M(3,1,2)]] || <math>kS_3</math> || 1 ||3 ||2 ||<math>C_2</math> ||<math>C_2</math> || <math>k^* \times C_2</math> ||1 ||1 || [[Image:M(3,1,2)tree.png|45px]] |
|} | |} | ||
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| + | [[Category:Cyclic p-group]] | ||
| + | [[Category: p-group]] | ||
Latest revision as of 10:20, 22 November 2018
Blocks with defect group [math]C_3[/math]
These are blocks with cyclic defect groups and so they are described by Brauer trees.
There are two [math]\mathcal{O}[/math]-Morita equivalence classes, accounting for all the possible Brauer trees.
| 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(3,1,1) | [math]kC_3[/math] | 1 | 3 | 1 | [math]1[/math] | [math]S_3[/math] | [math]k:k^*[/math] | 1 | 1 | tree.png){kind=small}
|
| M(3,1,2) | [math]kS_3[/math] | 1 | 3 | 2 | [math]C_2[/math] | [math]C_2[/math] | [math]k^* \times C_2[/math] | 1 | 1 | tree.png){kind=small}
|
