# C2xC2

## 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] |