13 - Loewy Series

Summary

The internal structure of indecomposable projective modules encodes, among other things, the possible extensions between simple modules (12.1). In Chapter 12 we looked closely at the Ext functor for finite groups of Lie type. Here we consider the broader question of describing the Loewy series of PIMs (13.1), which in principle can yield minimal projective resolutions of all simple modules (13.4).

Even though Loewy series for groups of Lie type have been worked out in detail only for SL(2, q) (see 13.5) and a few isolated groups of higher rank surveyed at the end of the chapter, the study of closely parallel theories in characteristic 0 (13.6) and characteristic p (13.7) suggests a natural candidate for the “generic” Loewy length of PIMs for KG when G is a finite group of Lie type over a field of p^r elements. This should be 2rm + 1, where m = |Φ⁺|.

Assuming the truth of Lusztig's Conjecture (3.11), Pillen exploits for finite Chevalley groups when r = 1 an embedding of principal series modules into Weyl modules (13.10) to prove that the generic Loewy length of PIMs is indeed uniform: 2m + 1 (13.11). This is in striking contrast to the fact that generic dimensions of PIMs typically vary from alcove to alcove for restricted weights. (Similar results are expected for r > 1 and for twisted groups, with generic Loewy length 2rm+1.)