Difference between revisions of "C9"
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== Blocks with defect group <math>C_9</math> == | == Blocks with defect group <math>C_9</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(9,1,1)]] || <math>kC_9</math> ||9 ||1 ||<math>1</math> ||<math>C_9 : C_6</math> || ||1 ||1 || | + | |[[M(9,1,1)]] || <math>kC_9</math> ||9 ||1 ||<math>1</math> ||<math>C_9 : C_6</math> || ||1 ||1 || [[Image:M(9,1,1)tree.png|45px]] |
|- | |- | ||
| − | |[[M(9,1,2)]] || <math>kD_{18}</math> ||6 ||2 || <math>C_2</math> || || ||1 ||1 || | + | |[[M(9,1,2)]] || <math>kD_{18}</math> ||6 ||2 || <math>C_2</math> || <math>C_2 \times C_3</math> || ||1 ||1 || [[Image:M(9,1,2)tree.png|45px]] |
|- | |- | ||
| − | |[[M(9,1,3)]] || <math>B_0( | + | |[[M(9,1,3)]] || <math>B_0(kSL_2(8))</math> ||6 ||2 ||<math>C_2</math> || <math>C_3</math> || ||1 ||1 || [[Image:M(9,1,3)tree.png|45px]] |
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Blocks in [[M(9,1,2)]] are derived equivalent (over <math>\mathcal{O}</math>) to those in [[M(9,1,3)]] | Blocks in [[M(9,1,2)]] are derived equivalent (over <math>\mathcal{O}</math>) to those in [[M(9,1,3)]] | ||
Latest revision as of 10:27, 22 November 2018
Blocks with defect group [math]C_9[/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(9,1,1) | [math]kC_9[/math] | 9 | 1 | [math]1[/math] | [math]C_9 : C_6[/math] | 1 | 1 | tree.png){kind=small}
| |
| M(9,1,2) | [math]kD_{18}[/math] | 6 | 2 | [math]C_2[/math] | [math]C_2 \times C_3[/math] | 1 | 1 | tree.png){kind=small}
| |
| M(9,1,3) | [math]B_0(kSL_2(8))[/math] | 6 | 2 | [math]C_2[/math] | [math]C_3[/math] | 1 | 1 | tree.png){kind=small}
|
Blocks in M(9,1,2) are derived equivalent (over [math]\mathcal{O}[/math]) to those in M(9,1,3)
