C2xC2 - Revision history

Charles Eaton: Added lifting column

2018-09-15T08:04:16Z

Added lifting column

Charles Eaton: NOTITLE

2018-09-09T09:18:19Z

__NOTITLE__

Charles Eaton at 11:37, 30 August 2018

2018-08-30T11:37:23Z

Charles Eaton at 13:33, 28 August 2018

2018-08-28T13:33:41Z

Charles Eaton at 12:54, 28 August 2018

2018-08-28T12:54:54Z

Charles Eaton: Tame blocks link added

2018-08-28T12:53:31Z

Tame blocks link added

Charles Eaton at 20:00, 23 August 2018

2018-08-23T20:00:49Z

Charles Eaton: Created page with "== Blocks with defect group $C_2 \times C_2$ == These are blocks were first classified over $k$ by Erdmann (see [Er82]]). Linckelmann classified them ov..."

2018-08-23T20:00:15Z

Created page with "== Blocks with defect group <math>C_2 \times C_2</math> == These are blocks were first classified over <math>k</math> by Erdmann (see [Er82]]). Linckelmann classified them ov..."

New page

== Blocks with defect group <math>C_2 \times C_2</math> ==

These are blocks were first classified over <math>k</math> by Erdmann (see [Er82]]). Linckelmann classified them over <math>\mathcal{O}</math> in [Li94], in which he also showed that the source algebras lie within three infinite families. In [CEKL11] the CFSG was used to show that only one source algebra can occur for each Morita equivalence class.

There are three <math>\mathcal{O}</math>-Morita equivalence classes.

{| 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(4,2,1)</math> || <math>k(C_2 \times C_2)</math> ||4 ||1 ||<math>1</math> ||<math>(C_2 \times C_2):S_3</math> || <math>(k \times k):GL_2(k)</math> ||1 ||1 ||
|-
|<math>M(4,2,2)</math> || <math>kA_4</math> ||4 ||3 ||<math>C_3</math> ||<math>S_3</math> || <math>(k^* \times k^* \times C_3):C_2</math> ||1 ||1 ||
|-
|<math>M(4,2,3)</math> || <math>B_0(kA_5)</math> ||4 ||3 ||<math>C_3</math> ||<math>C_2</math> || <math>(k^* \times k^*):C_2</math> ||1 ||1 ||
|}

@@ Line 4: / Line 4: @@
 These are blocks are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er82] ]]). Linckelmann classified them over <math>\mathcal{O}</math> in [[References|[Li94] ]], in which he also showed that the source algebras lie within three infinite families. In [[References|[CEKL11] ]] the CFSG was used to show that only one source algebra can occur for each Morita equivalence class.
 {| class="wikitable"
@@ Line 11: / Line 9: @@
 ! scope="col"| Class
 ! scope="col"| Representative
 ! scope="col"| <math>k(B)</math>
 ! scope="col"| <math>l(B)</math>
@@ Line 21: / Line 20: @@
 |-
-|[[M(4,2,1)|<math>M(4,2,1)</math>]] || <math>k(C_2 \times C_2)</math> ||4 ||1 ||<math>1</math> ||<math>(C_2 \times C_2):S_3</math> || <math>(k \times k):GL_2(k)</math> ||1 ||1 ||
+|[[M(4,2,1)]] || <math>k(C_2 \times C_2)</math> || 1 ||4 ||1 ||<math>1</math> ||<math>(C_2 \times C_2):S_3</math> || <math>(k \times k):GL_2(k)</math> ||1 ||1 ||
 |-
-|[[M(4,2,2)|<math>M(4,2,2)</math>]] || <math>B_0(kA_5)</math> ||4 ||3 ||<math>C_3</math> ||<math>C_2</math> || <math>(k^* \times k^*):C_2</math> ||1 ||1 || <math>D(3 {\cal A})_1</math>
+|[[M(4,2,2)]] || <math>B_0(kA_5)</math> || 1 ||4 ||3 ||<math>C_3</math> ||<math>C_2</math> || <math>(k^* \times k^*):C_2</math> ||1 ||1 || <math>D(3 {\cal A})_1</math>
 |-
-|[[M(4,2,3)|<math>M(4,2,3)</math>]] || <math>kA_4</math> ||4 ||3 ||<math>C_3</math> ||<math>S_3</math> || <math>(k^* \times k^* \times C_3):C_2</math> ||1 ||1 || <math>D(3 {\cal K})</math>
+|[[M(4,2,3)]] || <math>kA_4</math> || 1 ||4 ||3 ||<math>C_3</math> ||<math>S_3</math> || <math>(k^* \times k^* \times C_3):C_2</math> ||1 ||1 || <math>D(3 {\cal K})</math>
 |}

← Older revision		Revision as of 09:18, 9 September 2018
Line 1:		Line 1:
		+	__NOTITLE__
		+
	== Blocks with defect group <math>C_2 \times C_2</math> ==		== Blocks with defect group <math>C_2 \times C_2</math> ==

@@ Line 19: / Line 19: @@
 |-
-|<math>M(4,2,1)</math> || <math>k(C_2 \times C_2)</math> ||4 ||1 ||<math>1</math> ||<math>(C_2 \times C_2):S_3</math> || <math>(k \times k):GL_2(k)</math> ||1 ||1 ||
+|[[M(4,2,1)|<math>M(4,2,1)</math>]] || <math>k(C_2 \times C_2)</math> ||4 ||1 ||<math>1</math> ||<math>(C_2 \times C_2):S_3</math> || <math>(k \times k):GL_2(k)</math> ||1 ||1 ||
 |-
-|<math>M(4,2,2)</math> || <math>B_0(kA_5)</math> ||4 ||3 ||<math>C_3</math> ||<math>C_2</math> || <math>(k^* \times k^*):C_2</math> ||1 ||1 || <math>D(3 {\cal A})_1</math>
+|[[M(4,2,2)|<math>M(4,2,2)</math>]] || <math>B_0(kA_5)</math> ||4 ||3 ||<math>C_3</math> ||<math>C_2</math> || <math>(k^* \times k^*):C_2</math> ||1 ||1 || <math>D(3 {\cal A})_1</math>
 |-
-|<math>M(4,2,3)</math> || <math>kA_4</math> ||4 ||3 ||<math>C_3</math> ||<math>S_3</math> || <math>(k^* \times k^* \times C_3):C_2</math> ||1 ||1 || <math>D(3 {\cal K})</math>
+|[[M(4,2,3)|<math>M(4,2,3)</math>]] || <math>kA_4</math> ||4 ||3 ||<math>C_3</math> ||<math>S_3</math> || <math>(k^* \times k^* \times C_3):C_2</math> ||1 ||1 || <math>D(3 {\cal K})</math>
 |}

@@ Line 1: / Line 1: @@
 == Blocks with defect group <math>C_2 \times C_2</math> ==
-These are blocks are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [Er82]). Linckelmann classified them over <math>\mathcal{O}</math> in [Li94], in which he also showed that the source algebras lie within three infinite families. In [CEKL11] the CFSG was used to show that only one source algebra can occur for each Morita equivalence class.
+These are blocks are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er82] ]]). Linckelmann classified them over <math>\mathcal{O}</math> in [[References|[Li94] ]], in which he also showed that the source algebras lie within three infinite families. In [[References|[CEKL11] ]] the CFSG was used to show that only one source algebra can occur for each Morita equivalence class.
 There are three <math>\mathcal{O}</math>-Morita equivalence classes.

@@ Line 1: / Line 1: @@
 == Blocks with defect group <math>C_2 \times C_2</math> ==
-These are blocks were first classified over <math>k</math> by Erdmann (see [Er82]). Linckelmann classified them over <math>\mathcal{O}</math> in [Li94], in which he also showed that the source algebras lie within three infinite families. In [CEKL11] the CFSG was used to show that only one source algebra can occur for each Morita equivalence class.
+These are blocks are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [Er82]). Linckelmann classified them over <math>\mathcal{O}</math> in [Li94], in which he also showed that the source algebras lie within three infinite families. In [CEKL11] the CFSG was used to show that only one source algebra can occur for each Morita equivalence class.
 There are three <math>\mathcal{O}</math>-Morita equivalence classes.