A new proof of Stanley's theorem on the strong Lefschetz property

TL;DR

This paper presents a new, linear algebra-based proof of Stanley's theorem that all standard graded artinian monomial complete intersection algebras over characteristic zero fields possess the strong Lefschetz property, extending to certain positive characteristics.

Contribution

It provides a novel proof technique for Stanley's theorem using only basic linear algebra, applicable in broader characteristic settings.

Findings

New proof of Stanley's theorem using linear algebra

Extension of the strong Lefschetz property to fields with characteristic greater than the socle degree

Simplification of the proof method for the strong Lefschetz property

Abstract

A standard graded artinian monomial complete intersection algebra $A=\Bbbk[x_1,x_2,\ldots,x_n]/(x_1^{a_1},x_2^{a_2},\ldots,x_n^{a_n})$, with $\Bbbk$ a field of characteristic zero, has the strong Lefschetz property due to Stanley in 1980. In this paper, we give a new proof for this result by using only the basic properties of linear algebra. Furthermore, our proof is still true in the case where the characteristic of $\Bbbk$ is greater than the socle degree of $A$, namely $a_1+a_2+\cdots+a_n - n$.