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) |
Block. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Block&oldid=17013