Jordan degree type for codimension three Gorenstein algebras of small Sperner number

2024/06/10 by Nancy Abdallah, Nasrin Altafi, Abdallah, Nancy +5
Mathematics · #13E10 (Primary) #13H10 #14C05 (Secondary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2406.06322

openalex publication_date 2024/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Jordan type P_A,ℓ of a linear form ℓ acting on a graded Artinian algebra A over a field \sf k is the partition describing the Jordan block decomposition of the multiplication map m_ℓ, which is nilpotent. The Jordan degree type \mathcal S_A,ℓ is a finer invariant, describing also the initial degrees of the simple submodules of A in a decomposition of A as \sf k[ℓ]-modules. The set of Jordan types of A or Jordan degree types (JDT) of A as ℓ varies, is an invariant of the algebra. This invariant has been studied for codimension two graded algebras. We here extend the previous results to certain codimension three graded Artinian Gorenstein (AG) algebras - those of small Sperner number. Given a Gorenstein sequence T - one possible for the Hilbert function of a codimension three AG algebra - the irreducible variety Gor(T) parametrizes all Gorenstein algebras of Hilbert function T. We here completely determine the JDT possible for all pairs (A,ℓ), A∈ Gor(T), for Gorenstein sequences T of the form T=(1,3,s^k,3,1) for Sperner number s=3,4,5 and arbitrary multiplicity k. For s=6 we delimit the prospective JDT, without verifying that each occurs.

Related

• A note on Jordan types and Jordan degree types
• The Jordan type of graded Artinian Gorenstein algebras
• Reducibility of a family of local Artinian Gorenstein algebras
• Jordan type of an Artinian algebra, a survey
• Limits of graded Gorenstein algebras of Hilbert function (1,3^k,1)