C3 - Revision history

Charles Eaton at 10:20, 22 November 2018

2018-11-22T10:20:21Z

Charles Eaton: Categories and Pic_k

2018-10-07T14:56:10Z

Categories and Pic_k

Charles Eaton at 16:16, 18 September 2018

2018-09-18T16:16:48Z

Charles Eaton: Added lifting column

2018-09-15T08:18:49Z

Added lifting column

Charles Eaton: /* Blocks with defect group C_3 */

2018-09-09T12:27:51Z

‎Blocks with defect group C_3

Charles Eaton at 07:17, 29 August 2018

2018-08-29T07:17:57Z

Charles Eaton: Created links to classes

2018-08-28T17:00:30Z

Created links to classes

Charles Eaton: Created page with "== Blocks with defect group $C_3$ == These are blocks with cyclic defect groups and so they are described by Brauer trees. There are two $\mathcal{O}..."$

2018-08-23T12:29:37Z

Created page with "== 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}..."

New page

== 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="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(3,1,1)</math> || <math>kC_3</math> ||3 ||1 ||<math>1</math> ||<math>S_3</math> || ||1 ||1 ||
|-
|<math>M(3,1,2)</math> || <math>kS_3</math> ||3 ||2 ||<math>C_2</math> ||<math>C_2</math> || ||1 ||1 ||
|}

@@ Line 22: / Line 22: @@
 |-
-|[[M(3,1,1)]] || <math>kC_3</math> || 1 ||3 ||1 ||<math>1</math> ||<math>S_3</math> || ||1 ||1 || [[Image:M(3,1,1)tree.png|45px]]
+|[[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>kS_3</math> || 1 ||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>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]]
 |}

@@ Line 22: / Line 22: @@
 |-
-|[[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(3,1,1)]] || <math>kC_3</math> || 1 ||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>]] || 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]]
+|[[M(3,1,2)]] || <math>kS_3</math> || 1 ||3 ||2 ||<math>C_2</math> ||<math>C_2</math> || ||1 ||1 || [[Image:M(3,1,2)tree.png|45px]]
 |}

@@ Line 21: / Line 21: @@
 |-
-|[[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 ||
+|[[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(3,1,2)|<math>M(3,1,2)</math>]] || <math>kS_3</math> ||3 ||2 ||<math>C_2</math> ||<math>C_2</math> || ||1 ||1 ||
+|[[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]]
 |}

← Older revision		Revision as of 07:17, 29 August 2018
Line 1:		Line 1:
		+	__NOTITLE__
		+
	== Blocks with defect group <math>C_3</math> ==		== Blocks with defect group <math>C_3</math> ==

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>C_3</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 are two <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the possible Brauer trees.

@@ 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(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]]
 |}

@@ Line 19: / Line 19: @@
 |-
-|<math>M(3,1,1)</math> || <math>kC_3</math> ||3 ||1 ||<math>1</math> ||<math>S_3</math> || ||1 ||1 ||
+|[[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 ||
 |-
-|<math>M(3,1,2)</math> || <math>kS_3</math> ||3 ||2 ||<math>C_2</math> ||<math>C_2</math> || ||1 ||1 ||
+|[[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 ||
 |}