Modular representation theory

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 divides the order of G. In other words, the number of elements of G is zero when considered as an element of K. Such a representation is known as a modular representation. An example of modular representation theory would be the study of representations of the cyclic group of two elements over Z₂, the field with two elements.

Modular representations are very different from when K is the complex numbers, or when the characteristic of K does not divide the order of G. In those 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 zero when treated as an element of K.

Example

Finding a representation of the cyclic group of two elements over Z₂ 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 Z₂, 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 modular case, 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 non-modular case every irreducible representation is a direct summand in the regular representation, implying that it is projective.

The group algebra in the modular case is an artinian ring so that general structural results apply. 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.