{"paper":{"title":"Asymptotic Bounds for the Size of Hom$(A,{\\rm GL}_n(q))$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.GR","authors_text":"Alec Gullon, Michael Bate","submitted_at":"2014-05-22T15:06:36Z","abstract_excerpt":"Fix an arbitrary finite group $A$ of order $a$, and let $X(n,q)$ denote the set of homomorphisms from $A$ to the finite general linear group ${\\rm GL}_n(q)$. The size of $X(n,q)$ is a polynomial in $q$. In this note it is shown that generically this polynomial has degree $n^2(1-a^{-1}) - \\epsilon_r$ and leading coefficient $m_r$, where $\\epsilon_r$ and $m_r$ are constants depending only on $r := n \\mod a$. We also present an algorithm for explicitly determining these constants."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.5785","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"}