The Gabriel-Roiter measures of the indecomposables in a regular component of the 3-Kronecker quiver

Ringel-Hall numbers and the Gabriel-Roiter submodules of simple homogeneous regular modules

Let k be an arbitrary field and Q an acyclic quiver of tame type. Consider the path algebra kQ and the category of finite dimensional right modules Mod kQ. In the first part of the paper we deduce that the Gabriel-Roiter inclusions in preprojective indecomposables and simple homogeneous regulars (with dimension the minimal radical vector) as well as their Gabriel-Roiter measures are field independent. Using this result we can prove in a more general setting a theorem by Bo Chen which states that the Gabriel-Roiter submodule of a simple homogeneous regular (with dimension the minimal radical ve…

Notes on the Gabriel-Roiter measure

These notes give an introduction to the Gabriel-Roiter measure of a finite dimensional algebra. They are based on a series of four lectures at the "Advanced School and Conference on Representation Theory and Related Topics" in Trieste (ICTP, January 2006).

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 ind(A)-->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.

The Gabriel-Roiter measures and representation type

Let $Λ$ be an Artin algebra. A GR segment of $Λ$ is a sequence of GR measures which is closed under direct successors and direct predecessors. The number of the GR segments was conjectured to relate to the representation type of $Λ$. In this paper, let $k$ be an algebraically closed field and $Λ$ be a finite-dimensional hereditary $k$-algebra. We show that $Λ$ admits infinitely many GR segments if and only if $Λ$ is of wild representation type. Thus the finiteness of the number of the GR segments might be an alternative characterization of the tameness of finite dimensional algebras over algeb…

Gabriel-Roiter measure, representation dimension and rejective chains

The Gabriel-Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel-Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel-Roiter measure $μ$ in an abelian length category $\mathcal{A}$, there exists an object $X'$ which depends on $X$ and $μ$, such that $Γ= \operatorname{End}_{\mathcal{A}}(X \oplus X')$ h…

The Gabriel-Roiter filtration of the Ziegler spectrum

Inclusion preserving maps from modules over an Artin algebra to complete partially ordered sets are studied. This yields a filtration of the Ziegler spectrum which is indexed by all Gabriel-Roiter measures. Another application is a compactness result for the set of subcategories of finitely presented modules that are closed under submodules.

Existence of a multiplicative basis for a finitely spaced module over an aggregate

By [R. Bautista, P. Gabriel, A.V Roiter., L. Salmeron, Representation-finite algebras and multiplicative basis. Invent. Math. 81 (1985) 217-285.], a finite-dimensional algebra having finitely many isoclasses of indecomposable representations admits a multiplicative basis. In Sections 4.10-4.12 of [P. Gabriel, A. V. Roiter, Representations of finite-dimensional algebras. Encyclopaedia of Math. Sci., vol. 73, Algebra 8, Springer-Verlag, 1992] an analogous hypothesis was formulated for finitely spaced modules over an aggregate. We prove this conjecture.

关于外三角长度范畴的第一 Brauer-Thrall 猜想

The first Brauer-Thrall conjecture for extriangulated length categories

本文在外三角长度范畴中引入关于 $Θ$ 的 Gabriel-Roiter 测度并推广 Gabriel 性质，在技术性条件下证明第一 Brauer-Thrall 猜想。

The Auslander-Reiten Components in the Rhombic Picture

For an indecomposable module $M$ over a path algebra of a quiver of type $\widetilde{\mathbb A}_n$, the Gabriel-Roiter measure gives rise to four new numerical invariants; we call them the multiplicity, and the initial, periodic and final parts. We describe how these invariants for $M$ and for its dual specify the position of $M$ in the Auslander-Reiten quiver of the algebra.