Cartan Invariants

It is safe to say that the theory of modular representations of finite groups is not a part of the average mathematician's toolkit. Matrix representations of finite groups over the complex field, and the resulting characters (traces of matrices), occur rather widely in both pure and applied mathematics. But replacing complex numbers by elements of a finite or other field of prime characteristic p may seem esoteric, especially when one sees the complications which ensue when p divides the group order. Modular representations do turn up, however, in subjects other than just finite group theory (e.g., algebraic topology of various flavors). One aspect of modular representation theory which can be explained fairly readily to non-specialists is the matrix of Carton invariants. This originates in work of E. Cartan and others around the turn of the century, concerning finite dimensional algebras (as we would now say). While we are mainly interested here in the group algebra of a finite group, it is best to begin with algebras in general. Then we shall specialize to group algebras, ultimately settling on some of the most interesting of finite groups—those of Lie type. While keeping the prerequisites to a minimum, we shall try to introduce the reader to some of the intriguing open questions about Cartan invariants for such groups.