Modular representation theory
From Wikipedia, the free encyclopedia
In mathematics, modular representation theory is the branch of representation theory that studies linear representations of finite group G over a field K such that the characteristic of K is non-zero. An example of modular representation theory would be the study of representations of the cyclic group of two elements over F2, the field with two elements.
If the characteristic of K does not divide the order of G then modular representations are similar to characteristic zero representations. In these cases, Maschke's theorem yields that every representation is a direct sum of irreducible representations. The key step in the proof of Maschke's theorem is to average over the elements of the group, which fails when the order of G is divisible by the characteristic of K. In this case, the modular case, the representation theory is quite different from the characteristic 0 case, called the ordinary representation case. In particular, representations need not be direct sums of irreducible representations, which is always true in the ordinary case.
Contents |
[edit] Example
Finding a representation of the cyclic group of two elements over F2 is equivalent to the problem of finding matrices whose square is the identity matrix. Over every field of characteristic other than 2, we can always find a basis such that the matrix can be written as a diagonal matrix with only 1 or −1 occurring on the diagonal, such as
Over F2, we can find many other possible matrices, such as
[edit] Ring theory interpretation
In terms of ring theory, the group algebra
- K[G]
is not a semisimple ring in the case when G is divisible by the characteristic of K, thus it will have a Jacobson radical that is non-zero. This also implies that there will exist finite-dimensional modules for the group algebra which are not projective modules. By contrast, in the characteristic 0 case every irreducible representation is a direct summand in the regular representation, implying that it is projective.
The group algebra is an Artinian ring. Modular representation theory was developed by Richard Brauer from about 1940 onwards to provide more detailed information linked to the structure of G. Such results are applied in group theory to problems not directly phrased in terms of representations.
[edit] Number of simple modules
In ordinary representation theory, the number of simple modules k(G) is equal to the number of conjugacy classes. In the modular case, the number l(G) of simple modules is equal to the number of conjugacy classes whose elements have order coprime to the relevant prime p, the so-called p-regular classes.
[edit] Structure of the group algebra
In modular representation theory, while Maschke's theorem does not hold, the group algebra can be decomposed somewhat. In characteristic zero, The group algebra K[G] breaks up as the direct sum of n copies of each irreducible representation, where n is the dimension of each representation. In the modular case, we can find a collection of two-sided ideals, called blocks, such that they intersect trivially, and their sum is equal to the whole algebra. Each simple module belongs to a block, since every simple module shows up in K[G], and so lies in one of the blocks. The block containing the trivial module is the principal block. Every indecomposable module belongs to a block as well, and so the composition factors of an indecomposable module must all come from the same block.
[edit] Projective modules
In ordinary representation theory, every indecomposable module is irreducible, and so every module is projective. However, the simple modules in positive characteristic are rarely projective. Indeed, if a simple module is projective, then it lies in a block on its own. The block is said to have 'defect 0', in this case. Generally, the projective modules are difficult to understand.
For a finite group, the projective indecomposable modules are in a one-to-one correspondence with the simple modules, by an easy bijection: the socle and top of each projective indecomposable is simple, and this establishes the correspondence. The composition factors of the projective modules can be calculated as follows:
Given the ordinary and modular characters of a particular finite group, one can express the ordinary characters as a linear combination of the modular characters, and so arrive at a decomposition of the ordinary character. The values are always non-negative integers. The integers involved can be placed in a matrix, with the ordinary characters assigned rows and the modular characters assigned columns. This is referred to as the decomposition matrix, and is frequently labelled D. It is customary to place the trivial ordinary and modular characters in the first row and column respectively. The product of D with its transpose results in the Cartan matrix; this matrix gives the composition factors of a particular projective module.
[edit] Defect groups
To each block of a finite group, a certain p-subgroup, the defect group, is attached. This defect group has a strong influence on the structure of the block. For example, if the defect group is trivial, then the block contains just one simple module, just one ordinary character, the ordinary and modular characters are identical, and the simple module is projective.
The next easiest case is when the defect group is a cyclic group. Then there are only finitely many indecomposable modules lying in the block, and the structure of such blocks is relatively easy to understand. In all other cases, there are infinitely many indecomposable modules lying in a block.
The non-cyclic blocks can be divided into two cases: tame and wild. The tame blocks have as a defect group a dihedral group, semidihedral group or quaternion group, and their structure has been broadly determined in a series of papers by Karin Erdmann. The wild blocks are those whose indecomposable modules are in some sense 'hard' to understand, and extremely difficult to classify, even in principle.
