The boundary of the irreducible components for invariant subspace varieties

Abstract

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.

Two Partial Orders for Standard Young Tableaux

“…For the converse, Example 2 in Section 6.2 shows that the condition that β \ γ be a vertical strip is necessary. We show in [6] that several other relations of geometric or of algebraic nature lie between the box and the dominance relations. If those two are equal, then all the relations coincide.…”

“…The following theorem is shown in [6]: Theorem 3.2. Suppose X, Y are LR-tableaux of the same type and of shape which is a rook strip.…”

“…These orders are defined combinatorially and are of importance in the theory of invariant subspaces of nilpotent linear operators. They control the geometry of varieties of invariant subspaces of nilpotent linear operators, as they describe the degeneration relation and the boundaries of the irreducible components, see [4,5,6]. Therefore, it is important to investigate properties of these orders.…”

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

“…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.…”

The socle tableau as a dual version of the Littlewood-Richardson tableau

“…Combinatorics. The Littlewood-Richardson (LR) coefficient occurs in a meaningful way in a variety of areas in algebra: for example as the structure constant of the product of Schur polynomials in the ring of symmetric functions; as the multiplicity of a given irreducible representation of the symmetric group in a decomposition of the tensor product of two others; or as the number of irreducible components of invariant subspace varieties [1,3,4,5,6].…”