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

Abstract

Formulas for the dimension vectors of all objects M in the category 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 , controlling the category 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 S(6) is introduced and determined (Theorem 2.8). This is a much finer invariant than the usual dimension vector.

Applications of Littlewood-Richardson tableaux to computing generic extension of semisimple invariant subspaces of nilpotent linear operators

“…there is no chance to get nice description of all objects up to isomorphism, see [1,24]. However, there are many results developing this theory, for example [8,2,23,24,25,6,7,18,17]. In particular, the papers [25,18,17] discuss important interrelations between the invariant subspaces and the categories cohX of coherent sheaves over weighted projective lines cohX and singularity theory.…”

“…then Γ = Γ {1,2,4,7} ∪Γ {3} ∪Γ {5,6} and Γ = Γ {1,2,4,7} ∪ Γ {3,5,6} . Moreover Γ {1,2,4,7} is the tableau of X = P (0)⊕P (3, 5, 6)⊕E (6) , Γ {3} is the tableau of P (4), Γ {5,6} is the tableau of P (2, 1)⊕E (2) and Γ {3,5,6} is the tableau of D((1, 2, 4), ())⊕E (2) . By lemma 6.1 there exists the following short exact sequence: 0 → P (4) → D((1, 2, 4), ()) → P (2) → 0, so we have also the short exact sequence:…”