{"paper":{"title":"Varieties of $G_r$-summands in Rational $G$-modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Paul Sobaje","submitted_at":"2016-05-20T12:47:42Z","abstract_excerpt":"Let $G$ be a simple simply connected algebraic group over an algebraically closed field $k$ of characteristic $p$, with $r$-th Frobenius kernel $G_r$. Let $M$ be a $G_r$-module and $V$ a rational $G$-module. We put a variety structure on the set of all $G_r$-summands of $V$ that are isomorphic to $M$, and study basic properties of these varieties. We give a few applications of this work to the representation theory of $G$, primarily in providing some sufficient conditions for when a $G_r$-module decomposition of $V$ can be extended to a $G$-module decomposition. In particular we are interested"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06330","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"}