Generic classifications by p-group class

This page will contain results for generic classes of p-groups. It is very much under construction so the list below is not complete.

Cyclic p-groups

Click here for background on blocks with cyclic defect groups.

Morita equivalence classes are labelled by Brauer trees, but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each k-Morita equivalence class corresponds to an unique [math]\mathcal{O}[/math]-Morita equivalence class.

For [math]p=2,3[/math] every appropriate Brauer tree is realised by a block and we can give generic descriptions.

[math]2[/math]-blocks with cyclic defect groups

[math]3[/math]-blocks with cyclic defect groups

Tame blocks

Erdmann classified algebras which are candidates for basic algebras of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [Er90] ) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field k at present.

Abelian 2-groups with 2-rank at most three

These have been classified in [WZZ18] and [EL18a] with respect to [math]\mathcal{O}[/math].

Suppose [math]D \cong C_{2^{n_1}} \times C_{2^{n_2}} \times C_{2^{n_3}}[/math] where [math]n_1 \geq n_2 \geq n_3 \geq 0[/math].

[math]n_1 \geq n_2 \geq n_3 \geq 0[/math]   [math]n_1,n_2,n_3[/math] 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 [math]n_1 \geq n_2 \geq n_3 \geq 0[/math] M([math]C_{2^{n_1}} \times C_{2^{n_2}} \times C_{2^{n_3}}[/math],1) [math]k(C_{2^{n_1}} \times C_{2^{n_2}} \times C_{2^{n_3}})[/math] 1 [math]2^{n_1+n_2+n_3}[/math] 1 [math]1[/math] 1 1

Abelian 2-groups

Donovan's conjecture holds for 2-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.