The Charney-Davis conjecture for simple thin polyominoes

Abstract

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

arXiv commentTo appear in Communications in Algebra