Over a field F of characteristic p, to every block of a group algebra FG is associated a p-group D called the defect group.  The deformation and representation theoretical properties of the block depend very much on the group D. Among blocks of defect group D, those for which the group D is normal in G play a privileged role.  When D is abelian, there is a conjecture of Broue that every block has a tilting to a block with normal defect group.

 

The quiver of a block of normal defect group D is a connected component  of a McKay graph of a p'-group.  The skeleton of a block is obtained by taking one primitive idempotent from each matrix block of the quotient by the radical. We show that the field F enters into the relations of the skeleton only in so far as it skews the commutation relations of FD.  If D is abelian, then the block with normal defect group has separable deformations which affect only the non-commutation relations. 

 

One question of interest in block theory today, following a paper by Benson and Kessar on quantum complete intersections, is when Galois conjugate non-principal blocks are Morita equivalent (have the same skeleton).  We give a construction involving extraspecial groups for which we can show that the non-principal blocks are Morita equivalent.