{"paper":{"title":"Oriented Involutions, Symmetric and Skew-Symmetric Elements in Group Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Cesar Polcino Milies, Edgar G. Goodaire","submitted_at":"2011-08-23T16:23:22Z","abstract_excerpt":"Let $G$ be a group with involution * and $\\sigma\\colon G\\to\\{\\pm1\\}$ a group homomorphism. The map $\\sharp$ that sends $\\alpha=\\sum\\alpha_gg$ in a group ring $RG$ to $\\alpha^{\\sharp}=\\sum\\sigma(g)\\alpha_gg^*$ is an involution of $RG$ called an \\emph{oriented group involution}. An element $\\alpha\\in RG$ is \\emph{symmetric} if $\\alpha^{\\sharp}=\\alpha$ and \\emph{skew-symmetric} if $\\alpha^{\\sharp}=-\\alpha$. The sets of symmetric and skew-symmetric elements have received a lot of attention in the special cases that * is the inverse map on $G$ and/or $\\sigma$ is identically 1, but not in general. I"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4648","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"}