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.
Contents
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
Click here for background on 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.
Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [KoLa20].
Abelian [math]2[/math]-groups with [math]2[/math]-rank at most three
These have been classified in [WZZ18] and [EL18a] with respect to [math]\mathcal{O}[/math]. The derived equivalences classes with respect to [math]\mathcal{O}[/math] are known.
Let [math]l,m,n \geq 1[/math] be distinct with [math]l,m \neq 1[/math]
| [math]rk_2(D) \leq 3[/math] | |||||
| [math]D[/math] | # [math](\mathcal{O}[/math]-)Morita classes | # [math]\mathcal{O}[/math]-Derived equiv classes | Always endopermutation source Morita equivalent? | References | Notes |
|---|---|---|---|---|---|
| [math]C_{2^n}[/math] | 1 | 1 | Yes | [Li96] | Blocks with cyclic defect groups |
| [math]C_2 \times C_2[/math] | 3 | 2 | Yes | [Li94] | |
| [math]C_{2^m} \times C_{2^m}[/math] | 2 | 2 | Yes | [EKKS14] | |
| [math]C_{2^m} \times C_{2^n}[/math] | 1 | 1 | Yes | [Pu88] | Nilpotent blocks |
| [math]C_2 \times C_2 \times C_2[/math] | 8 | 4 | [Ea16] | ||
| [math]C_{2^m} \times C_{2^m} \times C_{2^m}[/math] | 4 | 4 | Yes | [WZZ18] | |
| [math]C_{2^m} \times C_2 \times C_2[/math] | 3 | 2 | [EL18a] | ||
| [math]C_{2^m} \times C_{2^m} \times C_{2^n}[/math] | 2 | 2 | [WZZ18] | ||
| [math]C_{2^l} \times C_{2^m} \times C_{2^n}[/math] | 1 | 1 | Yes | [Pu88] | Nilpotent blocks |
Minimal nonabelian [math]2[/math]-groups
Blocks with defect groups which are minimal nonabelian [math]2[/math]-groups of the form [math]P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle[/math] are classified in [EKS12]. There are two [math]\mathcal{O}[/math]-Morita equivalence classes, with representatives [math]\mathcal{O}P[/math] and [math]\mathcal{O}(P:C_3)[/math].
For arbitrary minimal nonabelian [math]2[/math]-groups, by [Sa16] blocks with such defect groups and the same fusion system are isotypic.
Homocyclic [math]2[/math]-groups when inertial quotient contains a Singer cycle
A Singer cycle is an element of order [math]p^n-1[/math] in [math]\operatorname{Aut}({C_p}^n) \cong GL_n(p)[/math], and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of [math](C_p)^n[/math].
[math]2[/math]-blocks with homocyclic defect group [math] D \cong (C_{2^m})^n [/math] whose inertial quotient [math] \mathbb{E} [/math] contains a Singer cycle are classified in [McK19]. In this situation, [math]\mathbb{E}[/math] has the form [math]E:F [/math] where [math]E \cong C_{2^n-1}[/math] and [math]F[/math] is trivial, or a subgroup of [math]C_n[/math]. There are three [math] \mathcal{O} [/math]-Morita equivalence classes when [math]m=1, n =3 [/math]; two when [math]m=1 , n \neq 3 [/math]; and only one when [math]m\gt 1 [/math]. The three classes have representatives [math] \mathcal{O}J_1 [/math], which occurs only when [math]m=1, n=3[/math]; [math] \mathcal{O}(SL_2(2^n):F) [/math], which occurs only when [math]m=1 [/math]; and [math] \mathcal{O}(D : \mathbb{E})[/math].
The Morita equivalence between the block and the class representative is known to be basic, possibly except when [math]m=1,n=3[/math], since the Morita equivalences between the principal block of [math]{\rm \operatorname{Aut}(SL_2(8))}[/math] and the blocks [math]B_0(\mathcal{O}({}^2G_2(3^{2m+1})))[/math] are not known to be basic.
