Difference between revisions of "C2xC2"
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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. | 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. | ||
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{| class="wikitable" | {| class="wikitable" | ||
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! scope="col"| Class | ! scope="col"| Class | ||
! scope="col"| Representative | ! scope="col"| Representative | ||
| + | ! 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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|- | |- | ||
| − | |[[M(4,2,1) | + | |[[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) | + | |[[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) | + | |[[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] |
