Salvato in:
| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2606.00599 |
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Sommario:
- We establish a convolution formula for the characteristic polynomial of a finite geometric semilattice $M$: \[ χ(M,st)=\sum_{X\in \underline{M}} s^{r-{\rm rk}_{\underline{M}}(X)}χ(\underline{M}^X,t)\,χ(M_{(X)},s), \] 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 $\mathbb{F}_p$.