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.

There are three [math]\mathcal{O}[/math]-Morita equivalence classes.

Class	Representative	[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
[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]
[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]