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.
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Presented by Henning Krause.
Dedicated to Professor Daniel Simson on the occasion of his 75th birthday
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Dowbor, P., Meltzer, H. Dimension Vectors of Indecomposable Objects for Nilpotent Operators of Degree 6 with One Invariant Subspace. Algebr Represent Theor 22, 99–140 (2019). https://doi.org/10.1007/s10468-017-9759-z
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DOI: https://doi.org/10.1007/s10468-017-9759-z
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