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Formulas for the dimension vectors of all objects  M  in the category  $\mathcal {S}(\tilde {6})$ S ( 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  $\mathcal {S}(\tilde {6})$ S ( 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  $\mathcal {S}(\tilde {6})$ S ( 6 ~ ) is introduced and determined (Theorem 2.8). This is a much finer invariant than the usual dimension vector.

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