Modular Representation Theory - Blocks and The Structure of The Group Algebra

Blocks and The Structure of The Group Algebra

In modular representation theory, while Maschke's theorem does not hold when the characteristic divides the group order, the group algebra may be decomposed as the direct sum of a maximal collection of two-sided ideals known as blocks (when the field K has characteristic 0, or characteristic coprime to the group order, there is also such a decomposition of the group algebra K as a sum of blocks (one for each isomorphism type of simple module), but the situation is relatively transparent (at least when K is sufficiently large): each block is a full matrix algebra over K, the endomorphism ring of the vector space underlying the associated simple module).

To obtain the blocks, the identity element of the group G is decomposed as a sum of primitive idempotents in Z(R), the center of the group algebra over the maximal order R of F. The block corresponding to the primitive idempotent e is the two-sided ideal e R. For each indecomposable R-module, there is only one such primitive idempotent that does not annihilate it, and the module is said to belong to (or to be in) the corresponding block (in which case, all its composition factors also belong to that block). In particular, each simple module belongs to a unique block. Each ordinary irreducible character may also be assigned to a unique block according to its decomposition as a sum of irreducible Brauer characters. The block containing the trivial module is known as the principal block.