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We show that the ideal generated by maximal minors (i.e., $(k+1)$-minors) of a $(k+1) \times n$ Vandermonde matrix is radical and Cohen-Macaulay. Note that this ideal is generated by all Specht polynomials with shape $(n-k,1,...,1)$., Comment: 6 pages, simplified the proof of the main result. To appear in Math. Scand

We give a detailed proof for Gordan-Noether’s results in “Ueber die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet.” C. Lossen has written a paper in a similar direction as the present paper, but did not provide a proof for every result. In our paper, every result is proved. Furthermore, our paper is independent of Lossen’s paper and includes a considerable number of new observations.

We relate the Weyr structure of a square matrix B to that of the t × t block upper triangular matrix C that has B down the main diagonal and first superdiagonal, and zeros elsewhere. Of special interest is the case t = 2 and where C is the nth Sierpinski matrix B n , which is defined inductively by B 0 = 1 and B n = [ B n − 1 B n − 1 0 B n − 1 ] . This yields an easy derivation of the Weyr structure of B n as the binomial coefficients arranged in decreasing order. Earlier proofs of the Jordan analogue of this had often relied on deep theorems from such areas as algebraic geometry. The result has interesting consequences for commutative, finite-dimensional algebras.

Let $\mathcal {V}=\bigsqcup _{i=0}^n\mathcal {V}_i$ be the lattice of subspaces of the $n$-dimensional vector space over the finite field $\mathbb{F} _q$, and let $\mathcal {A}$ be the graded Gorenstein algebra defined over $\mathbb{Q} $ which has $\mathcal {V}$ as a $\mathbb{Q} $ basis. Let $F$ be the Macaulay dual generator for $\mathcal {A}$. We explicitly compute the Hessian determinant $|{\partial ^2F}/{\partial X_i \partial X_j}|$, evaluated at the point $X_1 = X_2 = \cdots = X_N=1$, and relate it to the determinant of the incidence matrix between $\mathcal {V}_1$ and $\mathcal {V}_{n-1}$. Our exploration is motivated by the fact that both of these matrices naturally arise in the study of the Sperner property of the lattice and the Lefschetz property for the graded Artinian Gorenstein algebra associated to it.

We prove the strong Lefschetz property for certain complete intersections defined by products of linear forms, using a characterization of the strong Lefschetz property in terms of central simple modules., Comment: 9 pages

Let $F$ be a homogeneous polynomial in $n$ variables of degree $d$ over a field $K$. Let $A(F)$ be the associated Artinian graded $K$-algebra. If $B \subset A(F)$ is a subalgebra of $A(F)$ which is Gorenstein with the same socle degree as $A(F)$, we describe the Macaulay dual generator for $B$ in terms of $F$. Furthermore when $n=d$, we give necessary and sufficient conditions on the polynomial $F$ for $A(F)$ to be a complete intersection.

A homogeneous ideal $I$ of a polynomial ring $S$ is said to have the Rees property if, for any homogeneous ideal $J \subset S $ which contains $I$, the number of generators of $J$ is smaller than or equal to that of $I$. A homogeneous ideal $I \subset S$ is said to be $\mathfrak m$-full if $\mathfrak mI:y=I$ for some $y \in \mathfrak m$, where $\mathfrak m$ is the graded maximal ideal of $S$. It was proved by one of the authors that $\mathfrak m$-full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not $\mathfrak m$-full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property., Comment: 8 pages

We compute the resultants for quadratic binomial complete intersections. As an application we show that any quadratic binomial complete intersection can have the set of square-free monomials as a vector space basis if the generators are put in a normal form., Comment: 14 pages

Let $A$ be a graded complete intersection over a field and $B$ the monomial complete intersection with the generators of the same degrees as $A$. The EGH conjecture says that if $I$ is a graded ideal in $A$, then there should be an ideal $J$ in $B$ such that $B/J$ and $A/I$ have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property., Comment: 7 pages

2 Preliminaries 5 2.1 A list of notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The equi-degree monomial complete intersection as a tensor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Young tableaux and Specht polynomials . . . . . . . . . . . . . . . . . . . . 6 2.4 Nilpotent matrices and Jordan bases . . . . . . . . . . . . . . . . . . . . . . 6 2.5 The strong Lefschetz property . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Let B be the Boolean lattice on an n-set with B=@?B"i the rank decomposition. Let M(n,i) be the incidence matrix between B"i and B"n"-"i. We obtain a recursive formula for the determinant of the matrix M(n,i).

We prove that any quadratic complete intersection with a certain action of the symmetric group has the strong Lefschetz property over a field of characteristic zero. Furthermore, we discuss under what conditions its ring of invariants by a Young subgroup is a homogeneous complete intersection with a standard grading.

Let A = bigoplus_{i >= 0} A_i be a standard graded Artinian K-algebra, where char K = 0. Then A has the Weak Lefschetz property if there is an element ell of degree 1 such that the multiplication times ell : A_i --> A_{i+1} has maximal rank, for every i, and A has the Strong Lefschetz property if times ell^d : A_i --> A_{i+d} has maximal rank for every i and d. The main results obtained in this paper are the following. 1) EVERY height three complete intersection has the Weak Lefschetz property. (Our method, surprisingly, uses rank two vector bundles on P^2 and the Grauert-Mulich theorem.) 2) We give a complete characterization (including a concrete construction) of the Hilbert functions that can occur for K-algebras with the Weak or Strong Lefschetz property (and the characterization is the same one). 3) We give a sharp bound on the graded Betti numbers (achieved by our construction) of Artinian K-algebras with the Weak or Strong Lefschetz property and fixed Hilbert function. This bound is again the same for both properties. Some Hilbert functions in fact FORCE the algebra to have the maximal Betti numbers. 4) EVERY Artinian ideal in K[x,y] possesses the Strong Lefschetz property. This is false in higher codimension., To appear in J. Algebra

We show that the class of completely m-full ideals coincides with the class of componentwise linear ideals in a polynomial ring over an infinite field., 12 pages

Let k k be a field of characteristic 0. If I ⊂ k [ x , y , z ] I \subset k[x,y,z] is a complete intersection generated by three homogeneous elements of degrees d 1 , d 2 , d 3 d_1,d_2,d_3 with 2 ≤ d 1 ≤ d 2 ≤ d 3 2 \le d_1 \le d_2 \le d_3 , then the reduction of I I by a general linear form is minimally generated by three elements if and only if d 3 ≤ d 1 + d 2 − 2 d_3 \le d_1+d_2-2 .

In this chapter we would like to discuss a generalization of Lefschetz elements for an Artinian local ring to study the Jordan decomposition of a general element. The point of departure for us is Theorem 5.1 due to D. Rees. Several results from Chap. 6 (e.g., stable ideals, Borel fixed ideals, gin(I), etc) are needed at a few points in Chap. 5.

The Lefschetz property originates in the Hard Lefschetz Theorem for compact Kahler manifolds, so it is natural that some results discussed in the former chapters have geometric backgrounds. For example, Corollary 4.17 on the flat extension can be understood from the cohomology ring of projective space bundles in a geometric setting.

The purpose of this chapter is to illustrate a role played by the SLP in connection with the theory of Artinian rings and the Schur–Weyl duality. We assume that the reader is familiar with commutative algebra but perhaps without knowledge of representation theory, but we are hopeful that the expert in representation theory may also find the following sections of interest.

The main result of this chapter is Theorem 4.10. This may be regarded as a generalization of Theorem 3.34 which states that the SLP is preserved by tensor products. Using the main theorem, we give some examples of complete intersections with the strong Lefschetz property.

In this chapter we discuss topics of invariant theory such as coinvariant algebras of reflection groups. In particular the coinvariant algebras of real reflection groups have the SLP, and the set of Lefschetz elements is explicitly determined in most cases.

In this chapter we define the k-Lefschetz properties by generalizing the Lefschetz properties. The k-Lefschetz properties give us a way of computing generic initial ideals and graded Betti numbers of Artinian graded K-algebras.

We show a number of properties of the commutator algebra of a nilpotent matrix over a field. In particular we determine the simple modules of the commutator algebra. Then the results are applied to prove that certain Artinian complete intersections have the strong Lefschetz property.

We give a necessary and sufficient condition for a standard graded Artinian ring of the form $K[x_{1},\ldots,x_{n}]/I$, where $I$ is an ${\mathfrak{m}}$-full ideal, to have the weak Lefschetz property in terms of graded Betti numbers. This is a generalization of a theorem of Wiebe for componentwise linear ideals. We also prove that the class of componentwise linear ideals and that of completely ${\mathfrak{m}}$-full ideals coincide in characteristic zero and in positive characteristic, with the assumption that $\mathrm{Gin}(I)$ w.r.t. the graded reverse lexicographic order is stable.

Introduction In his paper [10] the author investigated the structure of m-full ideals by analysing their syzygies and, as one special case, showed how the Betti numbers of Borel stable ideals over polynomial rings can be computed. The same result, among other things, was also obtained by Eliahou and Kervaire[l] by a different method. Let a be a Borel stable ideal in a polynomial ring R and let V{ be the ideal generated by i generic linear forms and let tn_i_l be the type of the ideal a+ VJVt over R/Vt. Then the Betti numbers of a are linear combinations of to,...,tn_l with certain binomial numbers as coefficients, and conversely from the numbers t0, ...,tn_1 the Betti numbers can be recovered (see [10], corollary 9 and proposition 4). The present paper grew out of the question of deciding what sequences t0, ...,tn_1 of integers can arise from a Borel stable ideal as above. Our goal of this paper is to prove Theorems 4l and 4-2 of Section 4, in which we show that if we make the restriction on the ideal that it be generated by monomials of a fixed degree then the sequence t0,..., tn_^ is precisely the same as what has been known as the Hilbert series of a graded Artin algebra. In an earlier paper [11] the author showed some properties of m-primary m-full ideals. In this present paper we need to generalize these results to ideals which are not necessarily m-primary. Most results of [11] can be generalized to general m-full ideals. The Oth local cohomology module Ua:mYa of R/a plays an important role. These are the contents of Section 1 and have independent interest. In Section 2 we summarize some combinatorial formulae derived from Macaulay's theorem which characterizes the Hilbert function of homogeneous algebras (Theorem 2-2). In Section 3 we consider m-full ideals o in regular local rings which satisfy the condition m (lo = ma. The result (Theorem 3-1) will be applied, in Section 4, to Borel stable ideals generated by monomials all of degree d to obtain the theorems mentioned above.

We give a characterization of the Lefschetz elements in Artinian Gorenstein rings over a field of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Gorenstein rings which do not have the strong Lefschetz property.

The concept of an m-full ideal was introduced and studied first by D. Rees (unpublished). In 1983, after having considered and discussed the concept with Professor Rees, the author showed some properties of these ideals in [11], and other authors also have obtained results related to them (cf. [5, 7]). The purpose of this paper is to seek syzygies of m-full ideals and try to analyze their structure. Let a be an m-full ideal, and ᾱ the reduction by a general element. Then it is possible to determine the number of basic syzygies of a in terms of ᾱ. We show that this leads to a method for obtaining a set of basic syzygies of a provided that one for ᾱ is known (Theorem 6). Moreover the entire structure of the syzygy module is known when it is reduced by a general element. It turns out that a/za is the direct sum of ᾱ and copies of the residue field (Corollary 7).

We define the strong Lefschetz property for finite graded modules over graded Artinian algebras whose grading is not necessarily standard. We show that most results which have been obtained for Artinian algebras with standard grading can be extended for non-standard grading. Our results on the strong Lefschetz property for non-standard grading can be used to prove that certain Artinian complete intersections with standard grading have the strong Lefschetz property., 24 pages, To appear in Journal of Algebra

Let A be a standard graded Artinian algebra over a field of characteristic zero and let z be a linear form in A. We define the central simple modules for each such pair (A, z). Assume that A is Gorenstein. Then we prove that A has the strong Lefschetz property if and only if there exists a linear form z in A such that all central simple modules of the pair (A,z) have the strong Lefschetz property. In the course of proof we need to extend the definition of the strong Lefschetz property to finite graded modules over graded Artinian algebra, which previously was defined only for standard graded Artinian algebras., Comment: 20 pages, To be published in Journal of Pure and Applied Algebra

The notations used in this paper without explicit mention are listed below. Here R is a positively graded Noetherian ring, a a homogeneous ideal of R, and f, g, …, h are homogeneous elements of R.

An ideal a of a local ring (R, m) is called m-full if am: y = a for some y in a certain faithfully flat extension of R. The definition is due to Rees (unpublished) and he had obtained some elementary results (also unpublished). The present paper concerns some basic properties of m-full ideals. One result is the characterization of m-fullness in terms of the minimal number of generators of ideal, generalizing his result in a low dimensional case (Theorem 2, § 2).

Let A = R/, where R is a regular local ring of arbitrary dimension and is an ideal of R. If A is a Gorenstein ring and if height = 2, it is easily proved that A is a complete intersection, i.e., is generated by two elements (Serre [5], Proposition 3). Hence Gorenstein rings which are not complete intersections are of embedding codimension at least three. An example of these rings is found in Bass’ paper [1] (p. 29). This is obtained as a quotient of a three dimensional regular local ring by an ideal which is generated by five elements, i.e., generated by a regular sequence plus two more elements. In this paper, suggested by this example, we prove that if A is a Gorenstein ring and if height = 3, then is minimally generated by an odd number of elements. If A has a greater codimension, presumably there is no such restriction on the minimal number of generators for , as will be conceived from the proof.

1Let A be an Artin local ring. The Dilworth number of A is defined to be the supremum of p(Z) for ideals Z in A, where p is the minimal number of generators. In my paper [4], I developed some methods to compute this number when A is homogeneously graded over a field k, and especially if A is Gorenstein and ch k = 0, I showed that it is the dominating case that the Dilworth number is attained by a power of the maximal ideal. In such a case we shall say that A has the Sperner property. In the present paper we consider Artin rings (A, m), not necessarily graded, and compare the Dilworth number of A and that of Gr,(A), and it will be shown that the main result of [4] is valid without the assumption that the rings are homogeneous. Our main result is Theorem 7, where we prove that “most” Artin Gorenstein rings have the Sperner property. (Actually we prove a stronger result, which says that it has the “weak Stanley property.“) To explain the term “most,” we represent an Artin Gorenstein ring A in the

Denote by p_k the k-th power sum symmetric polynomial n variables. The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular sequence. We consider then the following problem: describe the subsets n powersums forming a regular sequence. A necessary condition is that n! divides the product of the degrees of the elements. To find an easily verifiable sufficient condition turns out to be surprisingly difficult already in 3 variables. Given positive integers a, Comment: corrected a typo in Lemma 2.4(b)