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We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results in particular give a self-contained proof of Cohen-Macaulayness of certain $h$-equals sets, a result previously obtained by Etingof-Gorsky-Losev over the complex numbers using rational Cherednik algebras., 42 pages, 2 figures

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 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.

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

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 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

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