{"paper":{"title":"Perfect matchings, Fano planes, and orthogonal bases of type $E_8$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"R.M. Green, Tianyuan Xu","submitted_at":"2026-06-16T01:11:05Z","abstract_excerpt":"We use perfect matchings and labelled Fano planes to construct and study the $2025$ orthogonal bases of positive roots in the $E_8$ root system. The set of these bases forms a highly structured, Bruhat-like graded poset $(\\Omega, \\leq_Q)$ whose rank function can be computed from the cardinalities of so-called generalized Rothe diagrams. We give combinatorial characterizations of these diagrams in terms of matchings and Fano planes, and we explain how to compute the ranks of the elements of $\\Omega$ using suitable combinatorial statistics such as the weights of perfect matchings. We establish s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.17393","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/2606.17393/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"}