What is modular representation theory for groups good for?

Question

I am an absolute beginner in modular representation theory of finite groups. I know some things in representation theory in characteristic zero. My questions are regarding the main goals of this part of representation theory.

For example, does the modular representation theory give information back regarding the structure of the group algebra or of the group itself? I would like to see some specific examples where this theory can be applied. I know that it studies the blocks of the group algebra but I don't know too many things about this.

What are the other applications of the theory? I have seen there are some interesting things related with modular forms also discussed here on the forum. Besides that, what are the other applications of this theory?

Answer 1

As Geoff explains, modular representations of finite groups do play an important role in some of the standard group-theoretic developments (although they play less of a role in the classification of simple groups than some people had expected initially). In fact, I think Richard Brauer began to develop these ideas, including block theory, in the 1940s because he saw connections with structural questions about finite groups. By now of course the subject is also pursued just because of its intrinsic beauty, but it has applications.

As Dylan Wilson points out in his comment, some aspects of the modular theory played a key role for Quillen in his work on algebraic K-theory. Brauer's results in both ordinary and modular representation theory are essential to Quillen's proof of the Adams conjecture here. These ideas such as "Brauer lifting" are emphasized in Serre's lecture notes (later translated as Springer GTM 42).

I'd add that for finite groups of Lie type defined over fields of prime characteristic $p$, the modular theory relative to $p$ has been helpful at times in the study of ordinary (characteristic 0) representations of those groups. Moreover, it has implications for the rational representations in characteristic $p$ of ambient algebraic groups, due to the close connections between irreducible representations of the finite groups and algebraic groups (Curtis, Steinberg, ...)

In a different direction, the study of Galois representations (and $p$-adic representations in general) has often required information about the modular representations of the finite groups of Lie type which occur naturally as quotients when reduction mod $p$ is applied to rings of $p$-adic integers. All of this is complicated to explain, but is part of the broader study of certain matrix groups over local fields and their representations. Which in turn has implications for the unifying program laid out by Langlands.

Answer 2

I was going to write a comment, but I think I will make it an answer, which is far from comprehensive.

If one is interested in applications to the structure of finite groups, possibly the best-known example is the proof of Glauberman's Z*-theorem, which is an essential tool for the classification of finite simple groups ( I believe it was the most cited paper in Group Theory in the 1970's). Modular representation theory was an indispensable tool in its proof, as in the proof of the Brauer-Suzuki theorem (concerning finite groups with (generalized) quaternion Sylow 2-subgroups) before it ( though Glauberman later published a rather longer proof of the Brauer-Suzuki theorem which only used ordinary character theory). In general, the uses of modular representation theory which contributed towards the Classification of Finite Simple groups tended to be in groups whose Sylow $2$-subgroups did not contained large elementary Abelian Sylow $2$-subgroups ( ``large" usually meaning of order at least 8). So the classification of finite groups with generalized quaternion, dihedral, or semi-dihedral, Sylow $2$-subgroups made use of results from block theory, though in some cases, ordinary character-theoretic proofs eventually replaced the uses of block theory. Other ways in which modular representation theory has played a role in finite group theory include the Hall-Higman theorem, which certainly influenced J.G. Thompson's early work, including his Ph.D. thesis. However, restricting attention to immediate applications to the structure of finite groups gives an unrepresentative picture. Other areas where modular representations of finite groups play a role include Number Theory ( eg they arise in Wiles' proof of Fermat's Last Theorem), and areas such as Coding Theory, among many others.