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arXiv · 2203.03487

The Charney-Davis conjecture for simple thin polyominoes

Manoj Kummini·Dharm Veer
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Abstract

Let $\mathcal{P}$ be a simple thin polyomino and $\Bbbk$ a field. Let $R$ be the toric $\Bbbk$-algebra associated to $\mathcal{P}$. Write the Hilbert series of $R$ as $h_{R}(t)/(1-t)^{\dim(R)}$. We show that $$(-1)^{\left\lfloor{\frac{\mathrm{deg} h_R(t)}{2}}\right\rfloor}h_{R}(-1) \geq 0$$ if $R$ is Gorenstein. This shows that the Gorenstein rings associated to simple thin polyominoes satisfy the Charney-Davis conjecture.

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Manoj Kummini, Dharm Veer. 2022-03-07. The Charney-Davis conjecture for simple thin polyominoes. https://arxiv.org/abs/2203.03487

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