{"paper":{"title":"Word maps and surface relations in symmetric groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT","math.RT"],"primary_cat":"math.GR","authors_text":"Ewan Cassidy","submitted_at":"2026-08-03T13:33:45Z","abstract_excerpt":"We study the expected number of fixed points of a random permutation obtained via a word map, with surface group constraints imposed. For stable irreducible characters $\\chi$ of the symmetric group $S_{n}$ and with $R_{g}=[a_{1},b_{1}]\\dots[a_{g},b_{g}]$ and $w\\in F_{2g}$, we compute $\\mathbb{E}_{S_{n}^{2g}}\\left[\\chi\\left(R_{g}(h)\\right)\\#\\mathrm{fix}\\left(w(h)\\right)\\right]$. We show that, if $w$ is a shortest representative for the conjugacy class of $\\gamma\\in\\Gamma_{g}=\\left\\langle a_{1},b_{1},\\dots,a_{g},b_{g}:R_{g}\\right\\rangle$, then this expectation is $O\\left(1/\\dim\\chi\\right)$. As a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.02210","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/2608.02210/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"}