Modular Representation Theory

Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.

Within finite group theory, character-theoretic results proved by Richard Brauer using modular representation theory played an important role in early progress towards the classification of finite simple groups, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2 subgroups were too small in an appropriate sense. Also, a general result on embedding of elements of order 2 in finite groups called the Z* theorem, proved by George Glauberman using the theory developed by Brauer, was particularly useful in the classification program.

If the characteristic of K does not divide the order of G, then modular representations are completely reducible, as with ordinary (characteristic 0) representations, by virtue of Maschke's theorem. The proof of Maschke's theorem relies on being able to divide by the group order, which is not meaningful when the order of G is divisible by the characteristic of K. In that case, representations need not be completely reducible, unlike the ordinary (and the coprime characteristic) case. Much of the discussion below implicitly assumes that the field K is sufficiently large (for example, K algebraically closed suffices), otherwise some statements need refinement.