Operations on arc diagrams and degenerations for invariant subspaces of linear operators

Abstract

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.

Operations on arc diagrams and degenerations for invariant subspaces of linear operators. Part II

“…In this paper we generalize results presented in [3] and [4] where arc diagrams were applied to investigate the degeneration order for a certain class of invariant subspaces of nilpotent linear operators. We enlarge the set of arc-moves introduced in [3] and compare the partial order induced by these moves with the partial order given by degenerations in a variety associated with invariant subspaces of nilpotent linear operators.…”

“…Since each move either decreases the number of crossings or the number of poles, ≤ arc is a partial order. The arc-moves (A), (B), (C) and (D) were introduced in [3] and the arc-move (E) is new. Formally, an arc diagram is a finite set of arcs and poles in the Poincaré half plane.…”

“…The case where all three partitions are fixed has been studied in [3] and [4]. There, arc moves of types (A) -(D) suffice to describe the boundary of a V ∆ in V β α,γ .…”

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

“…For each pair (X, Y ) of indecomposable objects in S 1 (k) we determine in the table below the dimension of the k-space Hom S (X, Y ) of all S-homomorphisms from X to Y , see [19,Lemma 4] and [12].…”

“…The paper is motivated by results presented in [12,13], where there are investigated relationships between Littlewood-Richardson tableaux and geometric properties of invariant subspaces of nilpotent linear operators. It is observed there that these relationships are deep and interesting.…”