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arXiv · 2306.04393

The weak Lefschetz property of whiskered graphs

Susan M. Cooper·Sara Faridi·Thiago Holleben·Lisa Nicklasson·Adam Van Tuyl
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Abstract

We consider Artinian level algebras arising from the whiskering of a graph. Employing a result by Dao-Nair we show that multiplication by a general linear form has maximal rank in degrees 1 and $n-1$ when the characteristic is not two, where $n$ is the number of vertices in the graph. Moreover, the multiplication is injective in degrees $<n/2$ when the characteristic is zero, following a proof by Hausel. Our result in the characteristic zero case is optimal in the sense that there are whiskered graphs for which the multiplication maps in all intermediate degrees $n/2,\ldots,n-2$ of the associated Artinian algebras fail to have maximal rank, and consequently, the weak Lefschetz property.

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Susan M. Cooper, Sara Faridi, Thiago Holleben, Lisa Nicklasson, Adam Van Tuyl. 2023-10-02. The weak Lefschetz property of whiskered graphs. https://arxiv.org/abs/2306.04393

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