{"paper":{"title":"An LR pair that can be extended to an LR triple","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Kazumasa Nomura","submitted_at":"2015-12-11T22:19:49Z","abstract_excerpt":"Fix an integer $d \\geq 0$, a field $\\mathbb{F}$, and a vector space $V$ over $\\mathbb{F}$ with dimension $d+1$. By a decomposition of $V$ we mean a sequence $\\{V_i\\}_{i=0}^d$ of $1$-dimensional $\\mathbb{F}$-subspaces of $V$ such that $V = \\sum_{i=0}^d V_i$ (direct sum). Consider $\\mathbb{F}$-linear transformations $A$, $B$ from $V$ to $V$. Then $A,B$ is called an LR pair whenever there exists a decomposition $\\{V_i\\}_{i=0}^d$ of $V$ such that $A V_i = V_{i-1}$ and $B V_i = V_{i+1}$ for $0 \\leq i \\leq d$, where $V_{-1}=0$ and $V_{d+1}=0$. By an LR triple we mean a $3$-tuple $A,B,C$ of $\\mathbb{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03840","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":""},"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"}