Difference between revisions of "C3"

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(Blocks with defect group C_3)
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== Blocks with defect group <math>C_3</math> ==
 
== Blocks with defect group <math>C_3</math> ==
  
These are blocks with [[cyclic defect groups]] and so they are described by [[Brauer trees]].
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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
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! 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>M(3,1,1)</math>]] || <math>kC_3</math> ||3 ||1 ||<math>1</math> ||<math>S_3</math> || ||1 ||1 || [[Image:M(3,1,1)tree.png|45px]]
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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>M(3,1,2)</math>]] || <math>kS_3</math> ||3 ||2 ||<math>C_2</math> ||<math>C_2</math> || ||1 ||1 || [[Image:M(3,1,2)tree.png|45px]]
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|[[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]]
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[[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 M(3,1,1)tree.png
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 M(3,1,2)tree.png