Block

An ideal of a ring is said to be indecomposable if, for any ideals and of , implies or . The ideal is called a direct summand of if for some ideal of . A block of is defined to be any ideal of which is an indecomposable direct summand of . By a block idempotent of one understands any primitive idempotent of the centre of (cf. also Centre of a ring). An ideal of is a block of if and only if for some (necessarily unique) block idempotent of . Thus blocks and block idempotents determine each other.

Any decomposition of of the form , where each is a block of , is called a block decomposition of . In general, such a decomposition need not exist, but it does exist if is semi-perfect (cf. Perfect ring). In the classical case where is semi-primitive Artinian (cf. Primitive ring; Artinian ring), each block of is a complete matrix ring over a suitable division ring, and the number of blocks of is equal to the number of non-isomorphic simple -modules.

The study of blocks is especially important in the context of group representation theory (see Representation of a group; [a1], [a2], [a3], [a4], [a5]). Here, the role of is played by the group algebra , where is a finite group and the commutative ring is assumed to be a complete Noetherian semi-local ring (cf. also Commutative ring; Noetherian ring; Local ring) such that has prime characteristic . The most important special cases are:

is a complete discrete valuation ring of characteristic with of prime characteristic ;

is a field of prime characteristic .

One of the most useful aspects of modular representation theory is the study of the distribution of the irreducible ordinary characters of into blocks. The main idea is due to R. Brauer and can be described as follows. Let be a finite group and let be a prime number. Assume that is a complete discrete valuation ring of characteristic , is the quotient field of and is of characteristic . Let be the set of all irreducible -characters of (cf. Character of a group) and write to indicate that is a block of whose corresponding block idempotent is , i.e., . The character is said to belong to the block of if (here is extended by -linearity to the mapping ). It turns out that if are all distinct blocks of , then is a disjoint union of the , , where denotes the set of irreducible -characters of belonging to . In the classical case studied by Brauer, namely when is a splitting field for , the irreducible -characters of are identifiable with the irreducible -characters of .

Assume that, in the context of the previous paragraph, is a splitting field for . Let be a block of and let be the order of Sylow -subgroups of (cf. Sylow subgroup). It turns out that there exists an integer , called the defect of , such that is the largest power of which divides for all . The notion of the defect of can be defined by purely ring-theoretic properties under much more general circumstances. Namely, it suffices to assume that is a complete Noetherian semi-local ring such that has prime characteristic (see [a5]; Noetherian ring).

For the classical case where is a splitting field for , one has the following famous problem, frequently called the Brauer -conjecture. Let be a block of and let . Is it true that , where is the defect of ? Although many special cases have been attacked successfully, the general case is still far from being solved. So far (1996), Brauer's -conjecture has not been verified for all finite simple groups. Moreover, it is not known whether Brauer's -conjecture can be reduced to the case of simple (or at least quasi-simple) groups (see [a5]).

References

[a1]	G. Karpilovsky, "Group representations" , 1 , North-Holland (1992)
[a2]	G. Karpilovsky, "Group representations" , 2 , North-Holland (1993)
[a3]	G. Karpilovsky, "Group representations" , 3 , North-Holland (1994)
[a4]	G. Karpilovsky, "Group representations" , 4 , North-Holland (1995)
[a5]	G. Karpilovsky, "Group representations" , 5 , North-Holland (1996)