{"paper":{"title":"On Matrix Product Factorization in Association Schemes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Allen W. Herman, Bobby Miraftab","submitted_at":"2026-07-16T11:16:08Z","abstract_excerpt":"We study matrix product factorizations (MPFs) in symmetric association schemes: identities $A_SA_T=A_U$ where $A_S,A_T,A_U$ are loopless unions of basic relations and the ordinary matrix product is again a $0$-$1$ adjacency matrix. We give equivalent structural and spectral criteria for MPFs, derive valency and rank restrictions, and analyze several standard families. For $2$-class schemes, the only nontrivial loopless MPF comes from the scheme of the $5$-cycle. For $P$-polynomial schemes, the distance-regular recurrence gives strong restrictions on products $A_1A_i$. We also prove a universal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.14848","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/2607.14848/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"}