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Algebras and Representation Theory

, Volume 22, Issue 1, pp 99–140 | Cite as

Dimension Vectors of Indecomposable Objects for Nilpotent Operators of Degree 6 with One Invariant Subspace

  • Piotr DowborEmail author
  • Hagen Meltzer
Open Access
Article
  • 57 Downloads

Abstract

Formulas for the dimension vectors of all objects M in the category \(\mathcal {S}(\tilde {6})\) of nilpotent operators with nilpotency degree bounded by 6, acting on finite dimensional vector spaces with invariant subspaces in a graded sense, are given (Theorem 2.3). For this purpose we realize a tubular algebra Λ, controlling the category \(\mathcal {S}(\tilde {6})\), as an endomorphism algebra of a suitable tilting bundle over a weighted projective line of type (2,3,6) (Theorem 3.6). Using this description and a concept of mono-epi type, the interval multiplicity vector of an object in \(\mathcal {S}(\tilde {6})\) is introduced and determined (Theorem 2.8). This is a much finer invariant than the usual dimension vector.

Keywords

Nilpotent operator Invariant subspace Submodule category Tubular algebra Dimension vector Interval multiplicity vector Exceptional object Weighted projective line Coherent sheaf Tilting bundle Telescoping functor Mono-epi representation Mono-epi-type 

Mathematics Subject Classification (2010)

16G30 16G20 14F05 47A15 15A04 16D70 16G70 

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Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland
  2. 2.Institute of MathematicsSzczecin UniversitySzczecinPoland

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