The GR-segments for tame quivers

2010/04/20 by Chen, Bo
#16G20 #16G70 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1004.3472

Abstract

A GR-segment for an artin algebra is a sequence of Gabriel-Roiter measures, which is closed under direct predecessors and successors. The number of the GR-segments indexed by natural numbers ℕ and integers ℤ probably relates to the representation types of artin algebras. Let k be an algebraically closed field and Q be a tame quiver (of type \widetilde\mathbbA_n, \widetilde\mathbbD_n, \widetilde𝔼₆, \widetilde𝔼₇, or \widetilde𝔼₈). Let b be the number of the isomorphism classes of the exceptional quasi-simple modules over the path algebra Λ=kQ. We show that the number of the ℕ- and ℤ-indexed GR-segments in the central part for Q is bounded by b+1. Therefore, there are at most b+3 GR segments.

Related

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• The Gabriel-Roiter measures and representation type
• The Gabriel-Roiter measure for \widetilde\mathbbA_n II
• Regular modules with preprojective Gabriel-Roiter submodules over n-Kronecker quivers
• Geometric characterizations of the representation type of hereditary\n algebras and of canonical algebras