{"paper":{"title":"The Lie algebra $\\mathfrak{sl}_4(\\mathbb C)$ and the hypercubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Paul Terwilliger, William J. Martin","submitted_at":"2025-05-06T20:01:28Z","abstract_excerpt":"We describe a relationship between the Lie algebra $\\mathfrak{sl}_4(\\mathbb C)$ and the hypercube graphs. Consider the $\\mathbb C$-algebra $P$ of polynomials in four commuting variables. We turn $P$ into an $\\mathfrak{sl}_4(\\mathbb C)$-module on which each element of $\\mathfrak{sl}_4(\\mathbb C)$ acts as a derivation. Then $P$ becomes a direct sum of irreducible $\\mathfrak{sl}_4(\\mathbb C)$-modules\n  $P = \\sum_{N\\in \\mathbb N} P_N$, where $P_N$ is the $N$th homogeneous component of $P$. For $N\\in \\mathbb N$ we construct some additional $\\mathfrak{sl}_4(\\mathbb C)$-modules ${\\rm Fix}(G)$ and $T$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.03951","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/2505.03951/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"}