Block

An ideal $ I $ of a ring $ A $ is said to be indecomposable if, for any ideals $ X $ and $ Y $ of $ A $, $ I = X \oplus Y $ implies $ X = 0 $ or $ Y = 0 $. The ideal $ I $ is called a direct summand of $ A $ if $ A = I \oplus J $ for some ideal $ J $ of $ A $. A block of $ A $ is defined to be any ideal of $ A $ which is an indecomposable direct summand of $ A $. By a block idempotent of $ A $ one understands any primitive idempotent of the centre of $ A $( cf. also Centre of a ring). An ideal $ B $ of $ A $ is a block of $ A $ if and only if $ B = Ae $ for some (necessarily unique) block idempotent $ e $ of $ A $. Thus blocks and block idempotents determine each other.

Any decomposition of $ A $ of the form $ A = B _ {1} \oplus \dots \oplus B _ {n} $, where each $ B _ {i} $ is a block of $ A $, is called a block decomposition of $ A $. In general, such a decomposition need not exist, but it does exist if $ A $ is semi-perfect (cf. Perfect ring). In the classical case where $ A $ is semi-primitive Artinian (cf. Primitive ring; Artinian ring), each block of $ A $ is a complete matrix ring over a suitable division ring, and the number of blocks of $ A $ is equal to the number of non-isomorphic simple $ A $- 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 $ A $ is played by the group algebra $ RG $, where $ G $ is a finite group and the commutative ring $ R $ is assumed to be a complete Noetherian semi-local ring (cf. also Commutative ring; Noetherian ring; Local ring) such that $ R/J ( R ) $ has prime characteristic $ p $. The most important special cases are:

$ R $ is a complete discrete valuation ring of characteristic $ 0 $ with $ R/J ( R ) $ of prime characteristic $ p $;

$ R $ is a field of prime characteristic $ p $.

One of the most useful aspects of modular representation theory is the study of the distribution of the irreducible ordinary characters of $ G $ into blocks. The main idea is due to R. Brauer and can be described as follows. Let $ G $ be a finite group and let $ p $ be a prime number. Assume that $ R $ is a complete discrete valuation ring of characteristic $ 0 $, $ K $ is the quotient field of $ R $ and $ R/J ( R ) $ is of characteristic $ p $. Let $ { \mathop{\rm Irr} } ( G ) $ be the set of all irreducible $ K $- characters of $ G $( cf. Character of a group) and write $ B = B ( e ) $ to indicate that $ B $ is a block of $ RG $ whose corresponding block idempotent is $ e $, i.e., $ B = RGe $. The character $ \chi \in { \mathop{\rm Irr} } ( G ) $ is said to belong to the block $ B = B ( e ) $ of $ RG $ if $ \chi ( e ) \neq 0 $( here $ \chi $ is extended by $ K $- linearity to the mapping $ \chi : {KG } \rightarrow K $). It turns out that if $ B _ {1} \dots B _ {n} $ are all distinct blocks of $ RG $, then $ { \mathop{\rm Irr} } ( G ) $ is a disjoint union of the $ { \mathop{\rm Irr} } ( B _ {i} ) $, $ 1 \leq i \leq n $, where $ { \mathop{\rm Irr} } ( B _ {i} ) $ denotes the set of irreducible $ K $- characters of $ G $ belonging to $ B _ {i} $. In the classical case studied by Brauer, namely when $ K $ is a splitting field for $ G $, the irreducible $ K $- characters of $ G $ are identifiable with the irreducible $ \mathbf C $- characters of $ G $.

Assume that, in the context of the previous paragraph, $ K $ is a splitting field for $ G $. Let $ B $ be a block of $ RG $ and let $ p ^ {a} $ be the order of Sylow $ p $- subgroups of $ G $( cf. Sylow subgroup). It turns out that there exists an integer $ d \geq 0 $, called the defect of $ B $, such that $ p ^ {a - d } $ is the largest power of $ p $ which divides $ \chi ( 1 ) $ for all $ \chi \in { \mathop{\rm Irr} } ( B ) $. The notion of the defect of $ B $ can be defined by purely ring-theoretic properties under much more general circumstances. Namely, it suffices to assume that $ R $ is a complete Noetherian semi-local ring such that $ R/J ( R ) $ has prime characteristic $ p $( see [a5]; Noetherian ring).

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

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