Let C be a basic indecomposable hereditary K-coalgebra, where K is an arbitrary field. We investigate a technique for studying C and left C-comodules by means of the left valued Gabriel quiver of C, an associated Tits quadratic form and locally nilpotent representations of the Extspecies of C.

We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are algebraic groups acting on these varieties. We give dimensions of orbits of these actions. Moreover, a combinatorial characterization of the partial order given by degenerations is described.

In this manuscript we show that two partial orders defined on the set of Littlewood-Richardson fillings of shape β \ γ and content α are equivalent if β \ γ is a horizontal and vertical strip. In fact, we give two proofs for the equivalence of the box order and the dominance order for fillings. Both are algorithmic. The first of these proofs emphasizes links to the Bruhat order for the symmetric group and the second provides a more straightforward construction of box moves. This work is motivated by the known result that the equivalence of the two combinatorial orders leads to a description of the geometry of the representation space of invariant subspaces of nilpotent linear operators.MSC 2010: Primary: 05E10, Secondary: 47A15

Assume that K is an arbitrary field. Let (I,) be a two-peak poset of finite prinjective type and let KI be the incidence algebra of I. We study sincere posets I and sincere prinjective modules over KI. The complete set of all sincere two-peak posets of finite prinjective type is given in Theorem 3.1. Moreover, for each such poset I, a complete set of representatives of isomorphism classes of sincere indecomposable prinjective modules over KI is presented in Tables 8.1.

Given partitions α, β, γ, the short exact sequences 0 −→ Nα −→ N β −→ Nγ −→ 0 of nilpotent linear operators of Jordan types α, β, γ, respectively, define a constructible subset V β α,γ of an affine variety. Geometrically, the varieties V β α,γ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson tableau Γ of shape (α, β, γ) contributes one irreducible component V Γ . We consider the partial order Γ ≤ boundary Γ on LR-tableaux which is the transitive closure of the relation given by V Γ ∩ V Γ = ∅. In this paper we compare the boundary relation with partial orders given by algebraic, combinatorial and geometric conditions. It is known that in the case where the parts of α are at most two, all those partial orders are equivalent. We prove that those partial orders are also equivalent in the case where β \ γ is a horizontal and vertical strip. Moreover, we discuss how the orders differ in general.MSC 2010: 14L30, 16G20, 47A15.