Varieties of invariant subspaces given by Littlewood-Richardson tableau

Abstract

Given partitions α,β,γ, the short exact sequences 0→Nα→Nβ→Nγ→0 of nilpotent linear operators of Jordan types α,β,γ, respectively, define a constructible subset Vαβ,γ of an affine variety. Geometrically, the varieties Vαβ,γ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson tableaux Γ of shape (α,β,γ) contributes one irreducible component V¯¯¯¯Γ. We consider the partial order Γ≤∗closureΓ~ on LR-tableaux which is the transitive closure of the relation given by VΓ~∩VΓ¯¯¯¯¯¯≠0. In this paper we compare the closure-relation with partial orders given by algebraic, combinatorial and geometric conditions. In the case where the parts of α are at most two, all those partial orders are equivalent. We discuss how the orders differ in general.