Difference between revisions of "C2xC2"

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== Blocks with defect group <math>C_2 \times C_2</math> ==
 
== 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.
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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.
 
  
 
{| class="wikitable"
 
{| class="wikitable"
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! scope="col"| Class
 
! scope="col"| Class
 
! scope="col"| Representative
 
! scope="col"| Representative
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! scope="col"| # lifts / <math>\mathcal{O}</math>
 
! scope="col"| <math>k(B)</math>
 
! scope="col"| <math>k(B)</math>
 
! scope="col"| <math>l(B)</math>
 
! scope="col"| <math>l(B)</math>
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|<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 ||
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|[[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 ||
 
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|-
|<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 ||  
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|[[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>
 
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|<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 ||  
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|[[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>
 
|}
 
|}

Latest revision as of 08:04, 15 September 2018

Blocks with defect group [math]C_2 \times C_2[/math]

These are blocks are examples of 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.

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
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]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]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]