Semistandard Filtrations in Highest Weight Categories

This paper is dedicated to the memory of Donald Higman. Don was especially attracted to methods with some kind of fundamental simplicity but with sufficient substance to be useful and illuminating. This paper is written in the spirit of this lofty aspiration and also somewhat in the direction of Don’s early interests in homological algebra. The paper begins in Section 1 with a foundational discussion of a new notion, that of a semistandard filtration in a highest weight category C. The latter categories [CPS1] axiomatize features found in many Lie-theoretic module settings. For simplicity we stick to the case of a finite weight poset , to which many considerations reduce. Here, the irreducible objects L(λ) and projective indecomposable objects P(λ) are indexed by , and there are standard objects (λ) with head L(λ) and all other composition factors indexed by a smaller weight. The objects P(λ) have finite filtrations with sections as standard objects, the top section being (λ) and all others of the form (ν) with ν > λ. In particular, these conditions imply ExtC( (λ), (μ)) = 0 unless λ > μ. Thus, whenever an object M in C has a finite length filtration of subobjects 0 = F0 ⊆ F1 ⊆ · · · ⊆ Fn = M