In the context of decomposition matrices for Hecke algebras of finite Coxeter groups at a root of unity (such as the tables at the end of the book "Hecke algebras at a root of unity" by GeckJacon or in GeckPfeiffer), what is meant by the defect of a block? Is there a simple explanation of what the defect keeps track of? I understand that the blocks of defect 1 have a decomposition matrix that has 1's on and just below the diagonal and 0's else, and that the blocks of higher defect have a more complicated structure. But what is "defect" exactly?
1 Answer
Originally ( in the context of group algebras of finite groups, which background is necessary to put later generalizations in context), the defect of a block was defined by Brauer as an arithmetical quantity. If $F$ is an algebraically closed field of prime characteristic $p,$ and $G$ is a finite group whose Sylow $p$subgroup has order $p^{a}$, then the defect of a block $B$ of $FG$ was defined to be $d,$ where $p^{ad}$ is highest power of $p$ which divides all degrees of irreducible characters assigned to $B.$ To each block with defect $d,$ Brauer assigned a unique (up to conjugacy) $p$subgroup $D$ of $G$ of order $p^{d}$, called the defect group of $B.$ The simplest case is when $d=0$ and $D = 1$. That occurs if and only if the block $B$ is a full matrix algebra over $F,$ in which case there is a unique simple module in $B,$ which is projective, and there is a unique irreducible (complex) character assigned to $B.$ Loosely speaking, the size and structure of $D$ is a good measure of the complexity of the algebra structure of $B,$ so that the larger the integer $d$ is, the more complicated the structure of the algebra $B$ (and its module category) is likely to be.
The structure of blocks of defect $1$ ( of group algebras) was worked out by Brauer in the 1940s, and the structure of blocks with cyclic defect group (at least at the character level) was worked out by Dade in the mid 1960s, with substantial contributions by JG Thompson and JA Green. K. Erdmann made major contributions to understanding blocks with defect groups which were dihedral, semidihedral, or generalized quaternion.
Module theoretic notions, such as relative projectivity, were linked to arithmetic properties by people such as JA Green, W. Feit, JL Alperin (and D.G. Higman). So for example, every module in a block $B$ is relatively $D$projective, which means that it is a direct summand of an $FG$module induced from $D.$ This means that every indecomposable $B$module has a vertex (as defined by J.A. Green) contained (up to conjugacy) in $D.$
By specializing the block theory of certain Hecke algebras at wellchosen roots of unity, information about blocks for finite groups can be obtained ( eg specializing the theory for type A Hecke Algebras at $p$th roots of unity yields information about the $p$blocks of the symmetric groups). In general for other algebraic structures, blocks are usually obtained via some sort of linkage principle ( sometimes defined via characters, as Brauer and Feit did in the group case, sometimes via certain modules, as Brauer did). In these more general contexts, simple modules are usually directly linked if they both occur as composition factors of the same projective indecomposable module, and linked when the notion of directly linked is extended by transitivity. Again speaking loosely, the larger the defect ( or the bigger the defect group), the more complicated the Cartan matrix of the block is likely to be.
Incidentally, if $B$ has defect $d$, all elementary divisors of the Cartan matrix for $B$ are powers of $p,$ and $p^{d},$ which occurs with multiplicity $1$, is the largest. The analogous statements for blocks of Hecke Algebras are rather more subtle.

1$\begingroup$ I suppose I should say, to be more precise, that the more complicated the group structure of the defect group $D$ is, the more complicated the structure of the block algebra $B$ is likely to be: for example, if $D$ is cyclic, of whatever size, the structure of the block is quite well understood. $\endgroup$ Aug 25, 2014 at 13:42

$\begingroup$ A helpful essay on a sometimes confusing notion. (I made just a couple of minor corrections.) $\endgroup$ Aug 25, 2014 at 16:23

$\begingroup$ Thanks Jim. Brief and incomplete of necessity books could be written. I am trying to spot the differences! $\endgroup$ Aug 25, 2014 at 16:26

$\begingroup$ Thanks Geoff! That's a great answer and very informative. I've lost access to my account so can't accept your answer, but thanks. I'm still a little confused: is the defect in the Hecke algebra setting also defined in terms of the order of pgroups as with the modular rep theory defect described in your answer?, even when the regular number specifying the root of unity is not prime? Does it reduce to modular rep theory somehow, or is defect for blocks of Hecke algebras mostly some sort of analogy that only lines up with the modular rep theory of the underlying Coxeter group in special cases? $\endgroup$ Aug 25, 2014 at 19:37

1$\begingroup$ @CatO Minor: I do not consider myself an expert on the representation theory of Hecke Algebras: in type A, there is some explanation of what defect (group) should mean in a 2001 Inventiones paper of Kulshammer,Olsson and myself "Generalized blocks for Symmetric Groups", and this works for nonprime integers as well. JB Gramain has also written papers about defect groups for more general blocks. Other authors to look at are S. Donkin, A. Mathas, R. Dipper and many more. $\endgroup$ Aug 25, 2014 at 20:54