Difference between revisions of "C9"

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(Picard groups)
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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 || [[Image:M(9,1,1)tree.png|45px]]
 
|[[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 || [[Image:M(9,1,2)tree.png|45px]]
+
|[[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(kPSL_2(8))</math> ||6 ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(9,1,3)tree.png|45px]]
+
|[[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]]
 
|}
 
|}
  
 
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)]]

Revision as of 21:00, 27 September 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 M(9,1,1)tree.png
M(9,1,2) [math]kD_{18}[/math] 6 2 [math]C_2[/math] [math]C_2 \times C_3[/math] 1 1 M(9,1,2)tree.png
M(9,1,3) [math]B_0(kSL_2(8))[/math] 6 2 [math]C_2[/math] [math]C_3[/math] 1 1 M(9,1,3)tree.png

Blocks in M(9,1,2) are derived equivalent (over [math]\mathcal{O}[/math]) to those in M(9,1,3)