Convolution-type Identity for Characteristic Polynomials of Geometric Semilattices

We establish a convolution formula for the characteristic polynomial of a finite geometric semilattice $M$: \[ \chi(M,st)=\sum_{X\in \underline{M}} s^{r-{\rm rk}_{\underline{M}}(X)}\chi(\underline{M}^X,t)\,\chi(M_{(X)},s), \] where $\underline{M}$ 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 $\mathbb{F}_p$.