{"paper":{"title":"On homomorphisms from finite subgroups of $SU(2)$ to Langlands dual pairs of groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.GR"],"primary_cat":"math.RT","authors_text":"Yuji Tachikawa, Yuki Kojima","submitted_at":"2025-05-02T13:20:11Z","abstract_excerpt":"Let $N(\\Gamma,G)$ be the number of homomorphisms from $\\Gamma$ to $G$ up to conjugation by $G$. Physics of four-dimensional $\\mathcal{N}=4$ supersymmetric gauge theories predicts that $N(\\Gamma,G)=N(\\Gamma , \\tilde G)$ when $\\Gamma$ is a finite subgroup of $SU(2)$, $G$ is a connected compact simple Lie group and $\\tilde G$ is its Langlands dual. This statement is known to be true when $\\Gamma=\\mathbb{Z}_n$, but the statement for non-Abelian $\\Gamma$ is new, to the knowledge of the authors. To lend credence to this conjecture, we prove this equality in a couple of examples, namely $(G,\\tilde G)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.01253","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/2505.01253/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"}