{"paper":{"title":"Bricks over preprojective algebras and join-irreducible elements in Coxeter groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Sota Asai","submitted_at":"2017-12-22T05:47:08Z","abstract_excerpt":"A (semi)brick over an algebra $A$ is a module $S$ such that the endomorphism ring $\\operatorname{\\mathsf{End}}_A(S)$ is a (product of) division algebra. For each Dynkin diagram $\\Delta$, there is a bijection from the Coxeter group $W$ of type $\\Delta$ to the set of semibricks over the preprojective algebra $\\Pi$ of type $\\Delta$, which is restricted to a bijection from the set of join-irreducible elements of $W$ to the set of bricks over $\\Pi$. This paper is devoted to giving an explicit description of these bijections in the case $\\Delta=\\mathbb{A}_n$ or $\\mathbb{D}_n$. First, for each join-i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08311","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}