Classification by p-group

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
1	1	1	1	1	Yes	Yes	Yes
2	1	C_2	1	1	Yes	Yes	Yes
3	1	C_3	2	2	Yes	Yes	Yes
4	1	C_4	1	1	Yes	Yes	Yes
4	2	<math>C_2 \times C_2<\math>	3	3	Yes	Yes	Yes