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 $A=B_1\oplus \cdots \oplus B_n$, 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 $\mathrm{Irr}(G)$ 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 $\chi \in \mathrm{Irr}(G)$ is said to belong to the block $B=B(e)$ of $RG$ if $\chi (e)\ne 0$( here $\chi$ is extended by $K$- linearity to the mapping $\chi :KG\to K$). It turns out that if $B_1\dots B_n$ are all distinct blocks of $RG$, then $\mathrm{Irr}(G)$ is a disjoint union of the $\mathrm{Irr}(B_i)$, $1\le i\le n$, where $\mathrm{Irr}(B_i)$ 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 $d\ge 0$, called the defect of $B$, such that $p^{a-d}$ is the largest power of $p$ which divides $\chi (1)$ for all $\chi \in \mathrm{Irr}(B)$. 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 $k(B)=|\mathrm{Irr}(B)|$. Is it true that $k(B)\le p^d$, 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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