Modular representation theory

From Exampleproblems

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 representation theory is quite different from the characteristic 0 case, and in particular representations need not be sums of irreducible representations.

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

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 charactreristic 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.