Complete intersection Jordan types in height two

Abstract

We determine every Jordan type partition that occurs as the Jordan block decomposition for the multiplication map by a linear form in a height two homogeneous complete intersection (CI) Artinian algebra A over an algebraically closed field k of characteristic zero or large enough. We show that these CI Jordan type partitions are those satisfying specific numerical conditions; also, given the Hilbert function H(A), they are completely determined by which higher Hessians of A vanish at the point corresponding to the linear form. We also show new combinatorial results about such partitions, and in particular we give ways to construct them from a branch label or hook code, showing how branches are attached to a fundamental triangle to form the Ferrers diagram.

Jordan types for graded Artinian algebras in height two

“…As a consequence of the above Proposition, we recover a result of [AIK,Theorem 3.7] providing the number of complete intersection Jordan types P ∈ P(T ). Recall that a complete intersection Jordan type of diagonal lengths T is a partition P of diagonal lengths T such that κ(P ) = 2. where for i = 1, .…”

“…What can we say about the Jordan types? Some initial work has been done in higher dimensions by several: in particular the Jordan degree type is symmetric for graded Gorenstein ideals (see [AIK,Lemma 3.22], [H-W, §4.1] [CsGo, Lemma 4.6] and references cited there).…”

Free extensions and Lefschetz properties, with an application to rings of relative coinvariants

“…The virtue of our proof is that we do not need to assume the Hilbert functions are unimodal a priori. Of course unimodality of the Hilbert function is a posteriori a consequence of the strong Lefschetz property for a graded Artinian algebra 4 .…”

“…The Jordan type of multiplication by a generic linear form ℓ ∈ A 1 ⊂ m A and the Hilbert function of a graded A determine whether A is strong or weak Lefschetz, or neither. Several authors have provided examples of algebras that are weak Lefschetz but not strong Lefschetz, or have studied the non-strong Lefschetz locus of certain graded Artinian algebras -see [4,8,11,12,14,31,47], also [24,25] and references cited there. A portion of these articles have considered Lefschetz properties for graded Artinian algebras arising as coinvariant algebras of groups acting on polynomial rings, see for example [19,30,32,47].…”

“…Examples: (Example 3.7): H(R G (3,3,2) G (3,3,3) ) = (1, 1, 2, 1, 2, 1, 1) (Increasing m): H(R G (4,4,2) G (4,4,3) ) = (1, 1, 2, 1, 2, 1, 2, 1, 1) H(R G (5,5,2) G (5,5,3) ) = (1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1); (Increasing n): H(R G (3,3,3) G (3,3,4) ) = (1, 1, 1, 2, 1, 1, 2, 1, 1, 1) H(R G (3,3,4) G (3,3,5) ) = (1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1); (Both increasing): H(R G (4,4,4) G (4,4,5) ) = (1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1). Remark 3.10 (Symmetric decomposition).…”