The Jordan type of graded Artinian Gorenstein algebras

Abstract

We study the general Jordan type of standard graded Artinian Gorenstein algebras, it is a finer invariant than Weak and Strong Lefschetz properties for those algebras. We prove that their Jordan types are determined by the rank of certain Mixed Hessians. We give a description of the possible Jordan types for algebras of low socle degree and low codimension.

Complete intersection Jordan types in height two

“…, k + d − 2. 3 We note that when k = 1, this sequence contains two 0's. Attaching a branch of length zero at a position in ∆ d represents leaving a gap at the corresponding position of ∆ d .…”

“…We give a direct proof of the special case of [11, Theorem 3.30, Equation 3.35] for sequences T satisfying Equation (1.1).…”

“…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 , .…”

“…When no Hessian is zero, the Jordan type P is (4,2), the conjugate of T , and is strong Lefschetz. For the CI algebra R/(x 2 , y 3 ), the multiplication m y has partition P y = (3,3), and only h 0 is zero; the multiplication m x has partition P x = (2, 2, 2) and both h 0 , h 1 are zero, while m x+y has partition (4,2). For the CI algebra R/(xy, x 3 + y 3 ) the multiplication m x (or m y ) has partition (4, 1, 1), and only h 1 is zero.…”

“…at Levico, June 24-29, 2018. Participants in the working group were Nasrin Altafi, Rodrigo Gondim, Tony Iarrobino, Leila Khatami, Gleb Nenashev, Lisa Nicklasson, Marti Salat Molto, Giorgio Ottaviani, and Yong-Su Shin; all contributed to the discussion, which began with a presentation by Rodrigo Gondim of a diagram method of Barbara Costa and himself [3]. Subsequently, the authors resolved some of the questions presented at the conference, resulting in this paper.…”