Sperner property and finite-dimensional Gorenstein algebras associated to matroids

Abstract

We prove the Lefschetz property for a certain class of finitedimensional Gorenstein algebras associated to matroids. Our result implies the Sperner property of the vector space lattice. More generally, it is shown that the modular geometric lattice has the Sperner property. We also discuss the Gröbner fan of the defining ideal of our Gorenstein algebra.

Strictness of the log-concavity of generating polynomials of matroids

“…See e.g. [AOV,ALOVI,ALOVII,BH1,BH2,COSW,EH,MN,NY,Ya]. Let H f = ( ∂ ∂x i ∂ ∂x j f ) be the Hessian matrix of a polynomial f .…”

Complete intersection Jordan types in height two

“…We denote by R = k[x, y] the polynomial ring in two variables over k. We will consider Artinian Gorenstein (so by F.H.S. Macaulay's result complete intersection) algebras A = R/ Ann F , where F ∈ E = k[X, Y ] is the Macaulay dual generator of A. T. Maeno and J. Watanabe in 2009 introduced a method of using higher Hessians to determine the strong or weak Lefschetz properties of a graded Artinian algebra [16]; this was further developed and used by T. Maeno and Y. Numata [15] and by R. Gondim and colleagues [4,5,3]. The Hilbert function of a graded CI quotient A of R satisfies H(A) = T , a symmetric sequence of the form T = (1 0 , 2 1 , .…”

Determinants of incidence and Hessian matrices arising from the vector space lattice

“…(b) Since the maps in (a) and (b) are dual to each other by the theory of inverse systems, it follows that ×ℓ n : A 0 → A n is given by multiplication by the same integer as the map in (a). This algebra has been proven to have the strong Lefschetz property in [9].…”

“…We fix this bijection once and for all, so that the variable X i corresponds to the vector v i ∈ P n−1 F . We now outline the construction given in [9] of a graded Artinian Gorenstein algebra associated to the vector space lattice. This uses the theory of Macaulay inverse systems, which provides a correspondence between homogeneous polynomials in the ring R and graded Artinian Gorenstein quotient algebras of Q.…”

“…In Section 3 we carry out our computation of the determinant of the incidence matrix between the first level set and the (n − 1)st level set. In Section 4 we recall Maeno-Numata's construction of the graded Artinian Gorenstein algebra A associated with the vector space lattice as introduced in [9]. We explicitly describe the Hessian matrix of the Macaulay dual generator of A and we compute the Hessian determinant.…”