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A binary tree of complete intersections with the strong Lefschetz property

Tadahito Harima, Satoru Isogawa, Junzo Watanabe

Abstract

In this paper we give a new family of complete intersections which have the strong Lefschetz property. The family consists of (Artinian algebras defined by) ideals generated by power sum symmetric polynomials of consecutive degrees and of certain ideals naturally derived from them. This family has a structure of a binary tree and this observation is a key to prove that all members in it have the strong Lefschetz property.

A binary tree of complete intersections with the strong Lefschetz property

Abstract

In this paper we give a new family of complete intersections which have the strong Lefschetz property. The family consists of (Artinian algebras defined by) ideals generated by power sum symmetric polynomials of consecutive degrees and of certain ideals naturally derived from them. This family has a structure of a binary tree and this observation is a key to prove that all members in it have the strong Lefschetz property.
Paper Structure (5 sections, 12 theorems, 165 equations)

This paper contains 5 sections, 12 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Central simple modules and the strong Lefschetz property
  3. Complete intersections defined by power sums of consecutive degrees
  4. Complete intersections defined by power sums and elementary symmetric polynomials
  5. The strong Lefschetz property for complete intersections

Key Result

Theorem 2.4

Let $A$ be a standard graded Artinian Gorenstein $K$-algebra. Then the following conditions are equivalent:

Theorems & Definitions (34)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4: HW1, Theorem 1.2
  • Remark 2.5
  • Theorem 3.1
  • Remark 3.2: Newton's identity
  • Remark 3.3
  • Proposition 3.5
  • proof
  • ...and 24 more