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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2606.00599 |
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| _version_ | 1866910275846799360 |
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| author | Chen, Yanru Fu, Houshan Li, Ang Wang, Suijie |
| author_facet | Chen, Yanru Fu, Houshan Li, Ang Wang, Suijie |
| contents | 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 $\mathbb{F}_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_00599 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convolution-type Identity for Characteristic Polynomials of Geometric Semilattices Chen, Yanru Fu, Houshan Li, Ang Wang, Suijie Combinatorics 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 $\mathbb{F}_p$. |
| title | Convolution-type Identity for Characteristic Polynomials of Geometric Semilattices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2606.00599 |