Defect group of a block
Let be a commutative ring, let
be an
-algebra, and let
be a group. Then
is said to be
-algebra if
acts as a group of
-algebra automorphisms of
. Expressed otherwise, each
acts on each
to give
such that this
-action makes
into a left
-module and
for all
,
. Assume that
is a subgroup of
. It is customary to write
for the subalgebra of
consisting of all
for which
for all
. The group algebra
is a
-algebra. Indeed, for any
and
, define
. Then
becomes a
-algebra over
. Further examples of
-algebras can be found in [a5].
Let be a
-algebra over
, where
is a finite group, let
be subgroups of
, and let
be a left transversal for
in
(i.e.,
picks out one element in each (e.g., left) coset of
in
; thus, it is a transversal of the system of subsets consisting of the (left) cosets; cf. also Combinatorial analysis). Then the mapping
,
, is
-linear and is independent of the choice of
. It is customary to refer to
as the relative trace mapping and to denote the image of
by
. Let
be a primitive idempotent of
. A subgroup
of
is called a defect group of
if
is a minimal element in the set of subgroups
of
such that
. Such a subgroup
exists because the set of all subgroups
of
with
contains
, and hence is non-empty. In the special case where
it is clear that
is a block idempotent of
(cf. Block). For this reason the defect groups of
are also called the defect groups of the block
.
Again let be a finite group, let
be a
-algebra over
, and let
be a primitive idempotent of
. Denote by
a defect group of
and assume that
is semi-perfect (e.g.,
is a finitely generated
-module and
is a complete Noetherian semi-local ring, cf. also Noetherian ring; Local ring). If
is a subgroup of
such that
, then
for some
. Moreover, a subgroup
of
is a defect group of
if and only if
is
-conjugate to
. In the most important case where
(
a prime number),
turns out to be a
-subgroup of
(see [a5]).
Let be a complete Noetherian semi-local ring such that
(
a prime number) and let
be a block of
. Then a defect group of
is a
-subgroup of
. Thus, if
is a defect group of
, then
for some integer
. The integer
is called the defect of
. The study of defect groups of blocks of
is especially important in the following cases:
is a complete discrete valuation ring of characteristic
with
of prime characteristic
;
is a field of prime characteristic
(see [a1], [a2], [a3], [a4], [a5]).
The ring-theoretic structure of blocks of is closely related to the structure of their defect groups. The most extensive coverage of this topic can be found in [a5]. For various applications to the modular representation theory of groups, see also [a5]. There is no doubt that the current theory of defect groups of blocks will be significantly strengthened in the future, and that the field contains untapped reserves rich enough to ensure major strikes. For various open problems and the current (1996) state of the subject, 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) |
Defect group of a block. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Defect_group_of_a_block&oldid=46601