{"paper":{"title":"Signatures of Type $A$ Root Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hung Manh Tran, Michael Cuntz, Shuhei Tsujie, Tan Nhat Tran","submitted_at":"2025-04-07T18:45:50Z","abstract_excerpt":"Given a type $A$ root system $\\Phi$ of rank $n$, we introduce the concept of a signature for each subset $S$ of $\\Phi$ consisting of $n+1$ positive roots. For a subset $S$ represented by a tuple $(\\beta_1, \\ldots, \\beta_{n+1})$, the signature of $S$ is defined as an unordered pair $\\{a, b\\}$, where $a$ and $b$ denote the numbers of $1$s and $-1$s, respectively, among the cofactors $(-1)^k \\det(S \\setminus \\{\\beta_k\\})$ for $1 \\le k \\le n+1$. We prove that the number of tuples with a given signature can be expressed in terms of classical Eulerian numbers. The study of these signatures is motiva"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.05423","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/2504.05423/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"}