11 - Cartan Invariants

Summary

The Cartan invariants of a group algebra KG (or other finite dimensional algebra) are the multiplicities of simple modules as composition factors of the principal indecomposable modules. If a set Λ (ordered in some way) indexes the isomorphism classes of simple modules L_λ and their projective covers U_λ, the resulting |Λ| × |Λ| Cartan matrixC has entries c_λμ := [U_λ : L_μ]. (There should be no confusion with the use of “Cartan matrix” in the context of root systems.)

For a group of Lie type there are two overlapping questions: Can one actually compute the matrix C for a specific group or family of groups? Can one find meaningful patterns in the entries of C, such as uniformities for a Lie type when p is sufficiently large?

After reviewing some general facts about Cartan invariants for a group algebra (11.1–11.3), we shall discuss what is known in the case of finite groups of Lie type, with emphasis on “generic” behavior for large p (11.9–11.11). We then survey the known computations and exhibit some small explicit examples, updating the 1985 survey [228]. The chapter concludes with a look at some recent conjectures involving block invariants.

Cartan Invariants for Finite Groups

If G is any finite group, consider the simple KG-modules L_λ (indexed by some set Λ) and their projective covers U_λ. Denote the composition factor multiplicities by cλμ = [U_λ : L_μ]. These are the Cartan invariants, forming a |Λ| × |Λ| matrix C.