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)