Difference between revisions of "C3"

Revision as of 08:18, 15 September 2018

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
[math]M(3,1,1)[/math]	1	[math]kC_3[/math]	3	1	[math]1[/math]	[math]S_3[/math]		1	1
[math]M(3,1,2)[/math]	1	[math]kS_3[/math]	3	2	[math]C_2[/math]	[math]C_2[/math]		1	1

@@ Line 11: / Line 11: @@
 ! scope="col"| Class
 ! scope="col"| Representative
+! scope="col"| # lifts / <math>\mathcal{O}</math>
 ! scope="col"| <math>k(B)</math>
 ! scope="col"| <math>l(B)</math>
@@ Line 21: / Line 22: @@
 |-
-|[[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]]
+|[[M(3,1,1)|<math>M(3,1,1)</math>]] || 1 || <math>kC_3</math> ||3 ||1 ||<math>1</math> ||<math>S_3</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]]
+|[[M(3,1,2)|<math>M(3,1,2)</math>]] || 1 || <math>kS_3</math> ||3 ||2 ||<math>C_2</math> ||<math>C_2</math> || ||1 ||1 || [[Image:M(3,1,2)tree.png|45px]]
 |}