Modular representation theory

In mathematics, modular representation theory is the branch of the representation theory of a finite group G in which linear representations are considered over a field K such that the order n of g is zero in K. In other words, the characteristic of K is a prime number p dividing the number of elements of G. This case has features that are essentially different from the case where K is the complex number field. These are seen already for representations of the cyclic group with two elements over the field with two elements; i.e. the classification of m × m binary matrices which square to the identity matrix.

In simple terms, in modular representation theory it is not possible to 'average over group elements', because that implies dividing by n, which is 0 in the modular arithmetic of K. In terms of ring theory, the group algebra

K[G]

is not a semisimple ring in the modular case. It will have a [[Jacobson radical] that is non-zero. Another way to put it is that there will be finite-dimensional modules for the group algebra that are not projective modules. In the non-modular case every irreducible representation is a direct summand in the regular representation, showing it directly to be 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.