Defect group of a block

Let $ R $ be a commutative ring, let $ A $ be an $ R $- algebra, and let $ G $ be a group. Then $ A $ is said to be $ G $- algebra if $ G $ acts as a group of $ R $- algebra automorphisms of $ A $. Expressed otherwise, each $ g \in G $ acts on each $ a \in A $ to give $ ^ {g} a \in A $ such that this $ G $- action makes $ A $ into a left $ RG $- module and $ ^ {g} ( ab ) = {} ^ {g} a {} ^ {g} b $ for all $ a,b \in A $, $ g \in G $. Assume that $ H $ is a subgroup of $ G $. It is customary to write $ A ^ {H} $ for the subalgebra of $ A $ consisting of all $ a \in A $ for which $ ^ {h} a = a $ for all $ h \in H $. The group algebra $ RG $ is a $ G $- algebra. Indeed, for any $ x \in RG $ and $ g \in G $, define $ ^ {g} x = gxg ^ {- 1 } $. Then $ RG $ becomes a $ G $- algebra over $ R $. Further examples of $ G $- algebras can be found in [a5].

Let $ A $ be a $ G $- algebra over $ R $, where $ G $ is a finite group, let $ H \subseteq K $ be subgroups of $ G $, and let $ T $ be a left transversal for $ H $ in $ K $( i.e., $ T $ picks out one element in each (e.g., left) coset of $ H $ in $ K $; thus, it is a transversal of the system of subsets consisting of the (left) cosets; cf. also Combinatorial analysis). Then the mapping $ { { \mathop{\rm Tr} } _ {H} ^ {K} } : {A ^ {H} } \rightarrow {A ^ {K} } $, $ a \mapsto \sum _ {t \in T } {} ^ {t} a $, is $ R $- linear and is independent of the choice of $ T $. It is customary to refer to $ { \mathop{\rm Tr} } _ {H} ^ {K} $ as the relative trace mapping and to denote the image of $ { \mathop{\rm Tr} } _ {H} ^ {K} $ by $ A _ {H} ^ {K} $. Let $ e $ be a primitive idempotent of $ A ^ {G} $. A subgroup $ D $ of $ G $ is called a defect group of $ e $ if $ D $ is a minimal element in the set of subgroups $ H $ of $ G $ such that $ e \in A _ {H} ^ {G} $. Such a subgroup $ D $ exists because the set of all subgroups $ H $ of $ G $ with $ e \in A _ {H} ^ {G} $ contains $ G $, and hence is non-empty. In the special case where $ A = RG $ it is clear that $ e $ is a block idempotent of $ RG $( cf. Block). For this reason the defect groups of $ e $ are also called the defect groups of the block $ B = RGe $.

Again let $ G $ be a finite group, let $ A $ be a $ G $- algebra over $ R $, and let $ e $ be a primitive idempotent of $ A ^ {G} $. Denote by $ D $ a defect group of $ e $ and assume that $ A ^ {G} $ is semi-perfect (e.g., $ A $ is a finitely generated $ R $- module and $ R $ is a complete Noetherian semi-local ring, cf. also Noetherian ring; Local ring). If $ H $ is a subgroup of $ G $ such that $ e \in A _ {H} ^ {G} $, then $ D \subseteq gHg ^ {- 1 } $ for some $ g \in G $. Moreover, a subgroup $ K $ of $ G $ is a defect group of $ e $ if and only if $ K $ is $ G $- conjugate to $ D $. In the most important case where $ { \mathop{\rm char} } ( R/J ( R ) ) = p $( $ p $ a prime number), $ D $ turns out to be a $ p $- subgroup of $ G $( see [a5]).

Let $ R $ be a complete Noetherian semi-local ring such that $ { \mathop{\rm char} } ( R/J ( R ) ) = p $( $ p $ a prime number) and let $ B $ be a block of $ RG $. Then a defect group of $ B $ is a $ p $- subgroup of $ G $. Thus, if $ D $ is a defect group of $ B $, then $ | D | = p ^ {d} $ for some integer $ d \geq 0 $. The integer $ d $ is called the defect of $ B $. The study of defect groups of blocks of $ RG $ is especially important in the following cases:

$ 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 $( see [a1], [a2], [a3], [a4], [a5]).

The ring-theoretic structure of blocks of $ RG $ 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)