Convolution-type Identity for Characteristic Polynomials of Geometric Semilattices

We establish a convolution formula for the characteristic polynomial of a finite geometric semilattice $M$: $$\chi (M,st)=\sum _{X\in \underset{‾}{M}}{s}^{r-{\mathrm{r}\mathrm{k}}_{\underset{‾}{M}}(X)}\chi ({\underset{‾}{M}}^X,t)\chi (M_{(X)},s),$$ where $\underset{‾}{M}$ denotes the centralization of $M$, and $M_{(X)}$ denotes the localization at $X$. This generalizes a nice formula of Southerland, Southern, and Zhou, which is recovered at $s=1$. When specialized to hyperplane arrangements, the identity yields a new expansion closely related to Wang’s convolution formula. We further provide a combinatorial interpretation of the convolution formula using the finite field method over ${\mathbb{F}}_{p^2}$ and ${\mathbb{F}}_p$.