{"paper":{"title":"Convolution-type Identity for Characteristic Polynomials of Geometric Semilattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ang Li, Houshan Fu, Suijie Wang, Yanru Chen","submitted_at":"2026-05-30T07:54:28Z","abstract_excerpt":"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 interpreta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.00599","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.00599/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}