Difference between revisions of "C(3^n)"
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== Blocks with defect group <math>C_{3^n}</math> == | == Blocks with defect group <math>C_{3^n}</math> == | ||
| − | These are blocks with | + | These are [[blocks with cyclic defect groups]] and so they are described by [[Brauer trees]]. |
| − | For each <math>n</math> there are three <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the possible Brauer trees. | + | For each <math>n>1</math> there are three <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the [[Glossary#Possible Brauer tree|possible Brauer trees]]. For <math>n=1</math> there are just two Morita equivalence classes (see [[C3|<math>C_3</math>]]). |
| − | In the | + | In the following <math>q_n</math> is a prime power such that <math>(q_n+1)_3=3^n</math>. For example take <math>q_n=2^{3^{n-1}}</math>. |
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| − | |[[M(3^n,1,1)|M(<math>3^n</math>,1,1)]] || <math>kC_{3^n}</math> || <math>3^n</math> ||1 ||<math>1</math> ||<math>C_{3^n} : C_{2.3^{n-1}}</math> || ||1 ||1 || | + | |[[M(3^n,1,1)|M(<math>3^n</math>,1,1)]] || <math>kC_{3^n}</math> || <math>3^n</math> ||1 ||<math>1</math> ||<math>C_{3^n} : C_{2.3^{n-1}}</math> || ||1 ||1 || [[Image:M(3^n,1,1)tree.png|45px]] |
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| − | |[[M(3^n,1,2)|M(<math>3^n</math>,1,2)]] || <math>kD_{2.3^n}</math> ||<math>\frac{(3^n+3)}{2}</math> ||2 ||<math>C_2</math> |||| ||1 ||1 || | + | |[[M(3^n,1,2)|M(<math>3^n</math>,1,2)]] || <math>kD_{2.3^n}</math> ||<math>\frac{(3^n+3)}{2}</math> ||2 ||<math>C_2</math> |||| ||1 ||1 || [[Image:M(3^n,1,2)tree.png|45px]] |
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| − | |[[M(3^n,1,3)|M(<math>3^n</math>,1,3)]] || <math>B_0(kPSL_2(q_n))</math> ||<math>\frac{(3^n+3)}{2}</math> ||2 ||<math>C_2</math> || || ||1 ||1 || | + | |[[M(3^n,1,3)|M(<math>3^n</math>,1,3)]] || <math>B_0(kPSL_2(q_n))</math> ||<math>\frac{(3^n+3)}{2}</math> ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(3^n,1,3)tree.png|45px]] |
|} | |} | ||
Blocks in [[M(3^n,1,2)|M(<math>3^n</math>,1,2)]] are derived equivalent (over <math>\mathcal{O}</math>) to those in [[M(3^n,1,3)|M(<math>3^n</math>,1,3)]]. | Blocks in [[M(3^n,1,2)|M(<math>3^n</math>,1,2)]] are derived equivalent (over <math>\mathcal{O}</math>) to those in [[M(3^n,1,3)|M(<math>3^n</math>,1,3)]]. | ||
Latest revision as of 23:34, 2 January 2019
Blocks with defect group [math]C_{3^n}[/math]
These are blocks with cyclic defect groups and so they are described by Brauer trees.
For each [math]n\gt 1[/math] there are three [math]\mathcal{O}[/math]-Morita equivalence classes, accounting for all the possible Brauer trees. For [math]n=1[/math] there are just two Morita equivalence classes (see [math]C_3[/math]).
In the following [math]q_n[/math] is a prime power such that [math](q_n+1)_3=3^n[/math]. For example take [math]q_n=2^{3^{n-1}}[/math].
| 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([math]3^n[/math],1,1) | [math]kC_{3^n}[/math] | [math]3^n[/math] | 1 | [math]1[/math] | [math]C_{3^n} : C_{2.3^{n-1}}[/math] | 1 | 1 | tree.png){kind=small}
| |
| M([math]3^n[/math],1,2) | [math]kD_{2.3^n}[/math] | [math]\frac{(3^n+3)}{2}[/math] | 2 | [math]C_2[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M([math]3^n[/math],1,3) | [math]B_0(kPSL_2(q_n))[/math] | [math]\frac{(3^n+3)}{2}[/math] | 2 | [math]C_2[/math] | 1 | 1 | tree.png){kind=small}
|
Blocks in M([math]3^n[/math],1,2) are derived equivalent (over [math]\mathcal{O}[/math]) to those in M([math]3^n[/math],1,3).
