C4 - Revision history

Charles Eaton at 10:18, 22 November 2018

2018-11-22T10:18:36Z

Charles Eaton: Categories and Pic_k

2018-10-07T14:09:36Z

Categories and Pic_k

Charles Eaton: Added lifting column

2018-09-15T08:02:02Z

Added lifting column

Charles Eaton: Image added

2018-09-09T12:22:40Z

Image added

Charles Eaton at 14:44, 8 September 2018

2018-09-08T14:44:54Z

Charles Eaton: Added link

2018-08-30T11:28:56Z

Added link

Charles Eaton: Created page with " == Blocks with defect group $C_4$ == These are blocks with cyclic defect groups and so they are described by Brauer trees. There is a unique $\mathc..."$

2018-08-23T11:42:20Z

Created page with " == Blocks with defect group <math>C_4</math> == These are blocks with cyclic defect groups and so they are described by Brauer trees. There is a unique <math>\mathc..."

New page

== Blocks with defect group <math>C_4</math> ==

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

There is a unique <math>\mathcal{O}</math>-Morita equivalence class.

{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Representative
! scope="col"| <math>k(B)</math>
! scope="col"| <math>l(B)</math>
! scope="col"| Inertial quotients
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math>
! scope="col"| <math>{\rm Pic}_k(B)</math>
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math>
! scope="col"| <math>{\rm mf_k(B)}</math>
! scope="col"| Notes

|-
|<math>M(4,1,1)</math> || <math>kC_4</math> ||4 ||1 ||<math>1</math> ||<math>D_8</math> || ||1 ||1
|}

@@ Line 22: / Line 22: @@
 |-
-|[[M(4,1,1)]] || <math>kC_4</math> || 1 ||4 ||1 ||<math>1</math> ||<math>D_8</math> || ||1 ||1 || [[Image:M(4,1,1)tree.png|45px]]
+|[[M(4,1,1)]] || <math>kC_4</math> || 1 ||4 ||1 ||<math>1</math> ||<math>D_8</math> || <math>(k \times k):k^*</math> ||1 ||1 || [[Image:M(4,1,1)tree.png|45px]]
 |}

@@ Line 21: / Line 21: @@
 |-
-|[[M(4,1,1)]] || <math>kC_4</math> ||4 ||1 ||<math>1</math> ||<math>D_8</math> || ||1 ||1
+|[[M(4,1,1)]] || <math>kC_4</math> ||4 ||1 ||<math>1</math> ||<math>D_8</math> || ||1 ||1 || [[Image:M(4,1,1)tree.png|45px]]
 |}

@@ Line 1: / Line 1: @@
 == Blocks with defect group <math>C_4</math> ==
@@ Line 20: / Line 21: @@
 |-
-|[[M(4,1,1)|<math>M(4,1,1)</math>]] || <math>kC_4</math> ||4 ||1 ||<math>1</math> ||<math>D_8</math> || ||1 ||1
+|[[M(4,1,1)]] || <math>kC_4</math> ||4 ||1 ||<math>1</math> ||<math>D_8</math> || ||1 ||1
 |}

@@ Line 20: / Line 20: @@
 |-
-|<math>M(4,1,1)</math> || <math>kC_4</math> ||4 ||1 ||<math>1</math> ||<math>D_8</math> || ||1 ||1
+|[[M(4,1,1)|<math>M(4,1,1)</math>]] || <math>kC_4</math> ||4 ||1 ||<math>1</math> ||<math>D_8</math> || ||1 ||1
 |}

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>C_4</math> ==
-These are blocks with [[cyclic defect groups]] and so they are described by [[Brauer trees]].
+These are [[blocks with cyclic defect groups]] and so they are described by [[Brauer trees]].
 There is a unique <math>\mathcal{O}</math>-Morita equivalence class.

@@ Line 11: / Line 11: @@
 ! scope="col"| Class
 ! scope="col"| Representative
 ! scope="col"| <math>k(B)</math>
 ! scope="col"| <math>l(B)</math>
@@ Line 21: / Line 22: @@
 |-
-|[[M(4,1,1)]] || <math>kC_4</math> ||4 ||1 ||<math>1</math> ||<math>D_8</math> || ||1 ||1 || [[Image:M(4,1,1)tree.png|45px]]
+|[[M(4,1,1)]] || <math>kC_4</math> || 1 ||4 ||1 ||<math>1</math> ||<math>D_8</math> || ||1 ||1 || [[Image:M(4,1,1)tree.png|45px]]
 |}