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This paper initiates a systematic study for key properties of Artinian Gorenstein \(K\)-algebras having binomial Macaulay dual generators. In codimension 3, we demonstrate that all such algebras satisfy the strong Lefschetz property, can be constructed as a doubling of an appropriate 0-dimensional scheme in \(\mathbb{P}^2\), and we provide an explicit characterization of when they form a complete intersection. For arbitrary codimension, we establish sufficient conditions under which the weak Lefschetz property holds and show that these conditions are optimal., Comment: Comments are welcome

We construct new families of Artinian Gorenstein graded $K$-algebras of arbitrary codimension having binomial Macaulay dual generators and satisfying the weak or the strong Lefschetz property. This is a companion paper to \cite{ADFMMSV}, which studies codimension three algebras having binomial Macaulay dual generators in great depth, establishing in particular that they enjoy the strong Lefschetz property., Comment: Comments are welcome

The Jordan type $P_{A,\ell}$ of a linear form $\ell$ acting on a graded Artinian algebra $A$ over a field $\sf k$ is the partition describing the Jordan block decomposition of the multiplication map $m_\ell$, which is nilpotent. The Jordan degree type $\mathcal S_{A,\ell}$ is a finer invariant, describing also the initial degrees of the simple submodules of $A$ in a decomposition of $A$ as ${\sf k}[\ell]$-modules. The set of Jordan types of $A$ or Jordan degree types (JDT) of $A$ as $\ell$ 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 $\mathrm{Gor}(T)$ parametrizes all Gorenstein algebras of Hilbert function $T$. We here completely determine the JDT possible for all pairs $(A,\ell), A\in \mathrm{Gor}(T)$, 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.

The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that a connected sum of doublings is the doubling of a fiber product ring.

We consider Artinian algebras $A$ over a field $\mathsf{k}$, both graded and local algebras. The Lefschetz properties of graded Artinian algebras have been long studied, but more recently the Jordan type invariant of a pair $(\ell,A)$ where $\ell$ is an element of the maximal ideal of $A$, has been introduced. The Jordan type gives the sizes of the Jordan blocks for multiplication by $\ell$ on $A$, and it is a finer invariant than the pair $(\ell,A)$ being strong or weak Lefschetz. The Jordan degree type for a graded Artinian algebra adds to the Jordan type the initial degree of ``strings'' in the decomposition of $A$ as a $\mathsf{k}[\ell]$ module. We here give a brief survey of Jordan type for Artinian algebras, Jordan degree type for graded Artinian algebras, and related invariants for local Artinian algebras, with a focus on recent work and open problems., Comment: 25 pages, comments welcome

We classify the minimal number of generators of artinian equigenerated monomial ideals $I$ such that $\Bbbk[x_1,\ldots,x_n]/I$ is forced to have the weak Lefschetz property., Comment: 27 pages

We study Hilbert functions, Lefschetz properties, and Jordan type of Artinian Gorenstein algebras associated to Perazzo hypersurfaces in projective space. The main focus lies on Perazzo threefolds, for which we prove that the Hilbert functions are always unimodal. Further we prove that the Hilbert function determines whether the algebra is weak Lefschetz, and we characterize those Hilbert functions for which the weak Lefschetz property holds. By example, we verify that the Hilbert functions of Perazzo fourfolds are not always unimodal. In the particular case of Perazzo threefolds with the smallest possible Hilbert function, we give a description of the possible Jordan types for multiplication by any linear form., Comment: 23 pages

We discuss whether the Jordan degree type encodes \break more information about graded artinian Gorenstein algebras than the Jordan type for linear forms. We show that in codimension two, the Jordan type determines the Jordan degree type. We provide examples showing that this is no longer the case in higher codimensions.

Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of $A$, we show that $A$ is almost strong Lefschetz. This quite modest result is nevertheless arguably the most encompassing so far concerning the strong Lefschetz property for graded codimension three AG algebras.

We give a sharp lower bound for the Hilbert function in degree $d$ of artinian quotients $\Bbbk[x_1,\ldots,x_n]/I$ failing the Strong Lefschetz property, where $I$ is a monomial ideal generated in degree $d \geq 2$. We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.

We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determined by at most three parameters., Comment: 24 pages

We prove that a sequence $h$ of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the strong Lefschetz property if and only if it is an SI-sequence. This generalizes the result by T. Harima which characterizes the Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provide classes of Artinian Gorenstein algebras obtained from the ideal of points in $\mathbb{P}^n$ such that some of their higher Hessians have non-vanishing determinants. Consequently, we provide families of such algebras satisfying the SLP., Comment: 16 pages; minor revision; to appear in the Proc. of the AMS

We let $A=R/I$ be a standard graded Artinian algebra quotient of $R={\sf k}[x,y]$, the polynomial ring in two variables over a field ${\sf k}$ by an ideal $I$, and let $n$ be its vector space dimension. The Jordan type $P_\ell$ of a linear form $\ell\in A_1$ is the partition of $n$ determining the Jordan block decomposition of the multiplication on $A$ by $\ell$ -- which is nilpotent. The first three authors previously determined which partitions of $n=\dim_{\sf k}A$ may occur as the Jordan type for some linear form $\ell$ on a graded complete intersection Artinian quotient $A=R/(f,g)$ of $R$, and they counted the number of such partitions for each complete intersection Hilbert function $T$ arXiv:1810.00716.\par We here consider the family $\mathrm{G}_T$ of graded Artinian quotients $A=R/I$ of $R={\sf k}[x,y]$, having arbitrary Hilbert function $H(A)=T$. The Jordan cell $\mathbb V(E_P)$ corresponding to a partition $P$ having diagonal lengths $T$ is comprised of all ideals $I$ in $R$ whose initial ideal is the monomial ideal $E_P$ determined by $P$. These cells give a decomposition of the variety $\mathrm{G}_T$ into affine spaces. We determine the generic number $\kappa(P)$ of generators for the ideals in each cell $\mathbb V(E_P)$, generalizing a result of arXiv:1810.00716. In particular, we determine those partitions for which $\kappa(P)=\kappa(T)$, the generic number of generators for an ideal defining an algebra $A$ in $\mathrm{G}_T$. We also count the number of partitions $P$ of diagonal lengths $T$ having a given $\kappa(P)$. A main tool is a combinatorial and geometric result allowing us to split $T$ and any partition $P$ of diagonal lengths $T$ into simpler $T_i$ and partitions $P_i$, such that $\mathbb V(E_P)$ is the product of the cells $\mathbb V(E_{P_i})$, and $T_i$ is single-block: $\mathrm{G}_{T_i}$ is a Grassmannian., Comment: 53 pages, 20 figures, new title, and revised for clarity

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 $\sf k$ 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 graph., Comment: 54 pages, 19 figures

We determine a sharp lower bound for the Hilbert function in degree $d$ of a monomial algebra failing the weak Lefschetz property over a polynomial ring with $n$ variables and generated in degree $d$, for any $d\geq 2$ and $n\geq 3$. We consider artinian ideals in the polynomial ring with $n$ variables generated by homogeneous polynomials of degree $d$ invariant under an action of the cyclic group $\mathbb{Z}/d\mathbb{Z}$, for any $n\geq 3$ and any $d\geq 2$. We give a complete classification of such ideals in terms of the weak Lefschetz property depending on the action., Comment: 27 pages

We study the WLP and SLP of artinian monomial ideals in $S=\mathbb{K}[x_1,\dots ,x_n]$ via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of $S/I$ is linear for at least $n-2$ steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial ideals with almost linear resolutions.

Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a square-free monomial ideal $I$ contains no variable and some power of $I$ is componentwise linear, then $I$ satisfies gcd condition. For a square-free monomial ideal $I$ which contains no variable, we show that $S/I$ is a Golod ring provided that for some integer $s\geq 1$, the ideal $I^s$ has linear quotient with respect to a monomial order. We also provide a lower bound for some Betti numbers of powers of a square-free monomial ideal which is generated in a single degree., Comment: arXiv admin note: text overlap with arXiv:math/0307222 by other authors