An Axiomatic Characterization of the Gabriel-roiter Measure

Given an abelian length category A, the Gabriel-Roiter measure with respect to a length function l is characterized as a universal morphism indA → P of partially ordered sets. The map is defined on the isomorphism classes of indecomposable objects of A and is a suitable refinement of the length function l. In his proof of the first Brauer-Thrall conjecture [5], Roiter used an induction scheme which Gabriel formalized in his report on abelian length categories [1]. The first BrauerThrall conjecture asserts that every finite dimensional algebra of bounded representation type is of finite representation type. Ringel noticed (see the footnote on p. 91 of [1]) that the formalism of Gabriel and Roiter works equally well for studying the representations of algebras having unbounded representation type. We refer to recent work [2, 3, 4] for some beautiful applications. In this note we present an axiomatic characterization of the Gabriel-Roiter measure which reveals its combinatorial nature. Given a finite dimensional algebra Λ, the GabrielRoiter measure is characterized as a universal morphism indΛ → P of partially ordered sets. The map is defined on the isomorphism classes of finite dimensional indecomposable Λ-modules and is a suitable refinement of the length function indΛ → N which sends a module to its composition length. The first part of this paper is purely combinatorial and might be of independent interest. We study length functions λ : S → T on a fixed partially ordered set S. Such a length function takes its values in another partially ordered set T , for example T = N. We denote by Ch(T ) the set of finite chains in T , together with the lexicographic ordering. The map λ induces a new length function λ : S → Ch(T ), which we call chain length function because each value λ(x) measures the lengths λ(xi) of the elements xi occuring in some finite chain x1 < x2 < . . . < xn = x of x in S. We think of λ ∗ as a specific refinement of λ and provide an axiomatic characterization. It is interesting to observe that this construction can be iterated. Thus we may consider (λ), ((λ)), and so on. The second part of the paper discusses the Gabriel-Roiter measure for a fixed abelian length category A, for example the category of finite dimensional Λ-modules over some algebra Λ. For each length function l on A, we consider its restriction to the partially ordered set indA of isomorphism classes of indecomposable objects of A. Then the Gabriel-Roiter measure with respect to l is by definition the corresponding chain length function l. In particular, we obtain an axiomatic characterization of l and use it to reprove Gabriel’s main property of the Gabriel-Roiter measure. Note that we work with a slight generalization of Gabriel’s original definition. This enables us to characterize the injective objects of A as those objects where l takes maximal values for some 2000 Mathematics Subject Classification. Primary: 18E10; Secondary: 06A07, 16G10. 1