Difference between revisions of "C27"
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== Blocks with defect group <math>C_{27}</math> == | == Blocks with defect group <math>C_{27}</math> == | ||
| − | These are blocks with | + | These are [[blocks with cyclic defect groups]] and so they are described by [[Brauer trees]]. |
There are three <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the possible Brauer trees. | There are three <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the possible Brauer trees. | ||
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|[[M(27,1,1)]] || <math>kC_{27}</math> ||27 ||1 ||<math>1</math> ||<math>C_{27} : C_{18}</math> || ||1 ||1 || | |[[M(27,1,1)]] || <math>kC_{27}</math> ||27 ||1 ||<math>1</math> ||<math>C_{27} : C_{18}</math> || ||1 ||1 || | ||
|- | |- | ||
| − | |[[M(27,1,2)]] || <math>kD_{54}</math> ||15 ||2 ||<math>C_2</math> |||| ||1 ||1 || | + | |[[M(27,1,2)]] || <math>kD_{54}</math> ||15 ||2 ||<math>C_2</math> || <math>C_{18}</math> || ||1 ||1 || |
|- | |- | ||
|[[M(27,1,3)]] || <math>B_0(kPSL_2(53))</math> ||15 ||2 ||<math>C_2</math> || || ||1 ||1 || | |[[M(27,1,3)]] || <math>B_0(kPSL_2(53))</math> ||15 ||2 ||<math>C_2</math> || || ||1 ||1 || | ||
Latest revision as of 10:26, 22 November 2018
Blocks with defect group [math]C_{27}[/math]
These are blocks with cyclic defect groups and so they are described by Brauer trees.
There are three [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(27,1,1) | [math]kC_{27}[/math] | 27 | 1 | [math]1[/math] | [math]C_{27} : C_{18}[/math] | 1 | 1 | ||
| M(27,1,2) | [math]kD_{54}[/math] | 15 | 2 | [math]C_2[/math] | [math]C_{18}[/math] | 1 | 1 | ||
| M(27,1,3) | [math]B_0(kPSL_2(53))[/math] | 15 | 2 | [math]C_2[/math] | 1 | 1 |
Blocks in M(27,1,2) are derived equivalent (over [math]\mathcal{O}[/math]) to those in M(27,1,3).
