{"paper":{"title":"Rooted tree modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Amit Kuber, Suraj Mishra","submitted_at":"2025-08-10T17:14:06Z","abstract_excerpt":"A rooted tree module (RTM) $M:=M(T,F)$ over a zero-relation algebra $\\Lambda:=\\mathcal KQ/\\langle\\rho\\rangle$ over a field $\\mathcal K$ is given by the data of a quiver morphism $F:T\\to Q$ from a rooted tree $T$ (either with a source or a sink) taking paths in $T$ to paths in $Q$ not lying in $\\langle\\rho\\rangle$. When $\\mathrm{char}(\\mathcal K)\\neq2$, we provide a checkable combinatorial characterization of the indecomposability of the RTM $M$ in terms of non-existence of idempotent quiver morphisms $\\iota:T\\to T$ satisfying $F\\circ\\iota=F$ and $\\iota\\neq 1_T$. Further, we provide an iterativ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.07435","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/2508.07435/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"}