Difference between revisions of "C5"

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(Picard groups)
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|[[M(5,1,1)]] || <math>kC_5</math> ||5 ||1 ||<math>1</math> ||<math>C_5:C_4</math> || ||1 ||1 || [[Image:M(5,1,1)tree.png|45px]]
 
|[[M(5,1,1)]] || <math>kC_5</math> ||5 ||1 ||<math>1</math> ||<math>C_5:C_4</math> || ||1 ||1 || [[Image:M(5,1,1)tree.png|45px]]
 
|-
 
|-
|[[M(5,1,2)]] || <math>kD_{10}</math> ||4 ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(5,1,2)tree.png|45px]]
+
|[[M(5,1,2)]] || <math>kD_{10}</math> ||4 ||2 ||<math>C_2</math> || <math>C_2 \times C_2</math> || ||1 ||1 || [[Image:M(5,1,2)tree.png|45px]]
 
|-
 
|-
|[[M(5,1,3)]] || <math>B_0(kA_5)</math> ||4 ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(5,1,3)tree.png|45px]]
+
|[[M(5,1,3)]] || <math>B_0(kA_5)</math> ||4 ||2 ||<math>C_2</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(5,1,3)tree.png|45px]]
 
|-
 
|-
 
|[[M(5,1,4)]] || <math>k(C_5:C_4)</math> ||5 ||4 ||<math>C_4</math> ||<math>C_4</math> || ||1 ||1 || [[Image:M(5,1,4)tree.png|45px]]
 
|[[M(5,1,4)]] || <math>k(C_5:C_4)</math> ||5 ||4 ||<math>C_4</math> ||<math>C_4</math> || ||1 ||1 || [[Image:M(5,1,4)tree.png|45px]]
 
|-
 
|-
|[[M(5,1,5)]] || <math>B_0(kA_7)</math> ||5 ||4 ||<math>C_4</math> || || ||1 ||1 || [[Image:M(5,1,5)tree.png|45px]]
+
|[[M(5,1,5)]] || <math>B_0(kA_7)</math> ||5 ||4 ||<math>C_4</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(5,1,5)tree.png|45px]]
 
|-
 
|-
|[[M(5,1,6)]] || faithful block of <math>k(2.A_7)</math> ||5 ||4 ||<math>C_4</math> |||| ||1 ||1 || [[Image:M(5,1,6)tree.png|45px]]
+
|[[M(5,1,6)]] || faithful block of <math>k(2.A_7)</math> ||5 ||4 ||<math>C_4</math> || <math>1</math> || ||1 ||1 || [[Image:M(5,1,6)tree.png|45px]]
 
|}
 
|}

Revision as of 07:02, 28 September 2018

Blocks with defect group [math]C_5[/math]

These are blocks with cyclic defect groups and so they are described by Brauer trees.

There are six [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(5,1,1) [math]kC_5[/math] 5 1 [math]1[/math] [math]C_5:C_4[/math] 1 1 M(5,1,1)tree.png
M(5,1,2) [math]kD_{10}[/math] 4 2 [math]C_2[/math] [math]C_2 \times C_2[/math] 1 1 M(5,1,2)tree.png
M(5,1,3) [math]B_0(kA_5)[/math] 4 2 [math]C_2[/math] [math]C_2[/math] 1 1 M(5,1,3)tree.png
M(5,1,4) [math]k(C_5:C_4)[/math] 5 4 [math]C_4[/math] [math]C_4[/math] 1 1 M(5,1,4)tree.png
M(5,1,5) [math]B_0(kA_7)[/math] 5 4 [math]C_4[/math] [math]C_2[/math] 1 1 M(5,1,5)tree.png
M(5,1,6) faithful block of [math]k(2.A_7)[/math] 5 4 [math]C_4[/math] [math]1[/math] 1 1 M(5,1,6)tree.png