Difference between revisions of "Classification by p-group"
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M(x,y,z) is a class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class. | M(x,y,z) is a class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class. | ||
| − | Note that it is not known that the isomorphism class of a defect group is a Morita invariant, so it could be that M(x,y1,z1)=M(x,y2,z2) for some <math>(y1,z1) \neq (y2,z2)<\math>. | + | Note that it is not known that the isomorphism class of a defect group is a Morita invariant, so it could be that <math>M(x,y1,z1)=M(x,y2,z2)<\math> for some <math>(y1,z1) \neq (y2,z2)<\math>. |
Also, at present there is no known example of a k-Morita equivalence class of blocks which splits into more than one Morita equivalence class with respect to a complete discrete valuation ring. If such an example arises, then we will bring in more notation for classes with respect to the d.v.r. | Also, at present there is no known example of a k-Morita equivalence class of blocks which splits into more than one Morita equivalence class with respect to a complete discrete valuation ring. If such an example arises, then we will bring in more notation for classes with respect to the d.v.r. | ||
| + | |||
| + | {| class="wikitable" | ||
| + | |- | ||
| + | ! scope="col"| |D| | ||
| + | ! scope="col"| SmallGroup | ||
| + | ! scope="col"| Isotype | ||
| + | ! scope="col"| k-classes | ||
| + | ! scope="col"| O-classes | ||
| + | ! scope="col"| Complete w.r.t. k? | ||
| + | ! scope="col"| Complete w.r.t. O? | ||
| + | ! scope="col"| Broue? | ||
| + | ! scope="col"| Notes | ||
| + | |- | ||
| + | | 3 | ||
| + | | 1 | ||
| + | | C_3 | ||
| + | | 2 | ||
| + | | 2 | ||
| + | | Yes | ||
| + | | Yes | ||
| + | | Yes | ||
| + | |} | ||
Revision as of 22:46, 16 August 2018
Classification of Morita equivalences for blocks with a given defect group
We use the following notation for Morita equivalence classes of blocks of finite groups with respect to an algebraically closed field k.
M(x,y,z) is a class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class.
Note that it is not known that the isomorphism class of a defect group is a Morita invariant, so it could be that <math>M(x,y1,z1)=M(x,y2,z2)<\math> for some <math>(y1,z1) \neq (y2,z2)<\math>.
Also, at present there is no known example of a k-Morita equivalence class of blocks which splits into more than one Morita equivalence class with respect to a complete discrete valuation ring. If such an example arises, then we will bring in more notation for classes with respect to the d.v.r.
| |D| | SmallGroup | Isotype | k-classes | O-classes | Complete w.r.t. k? | Complete w.r.t. O? | Broue? | Notes |
|---|---|---|---|---|---|---|---|---|
| 3 | 1 | C_3 | 2 | 2 | Yes | Yes | Yes |
