{"paper":{"title":"On the decomposition numbers of $\\mathrm{SO}_8^+(2^f)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RT","authors_text":"Alessandro Paolini","submitted_at":"2017-10-16T02:42:53Z","abstract_excerpt":"Let $q=2^f$, and let $G=\\mathrm{SO}_8^+(q)$ and $U$ be a Sylow $2$-subgroup of $G$. We first describe the fusion of the conjugacy classes of $U$ in $G$. We then use this information to prove the unitriangularity of the $\\ell$-decomposition matrices of $G$ for all $\\ell \\ne 2$ by inducing certain irreducible characters of $U$ to $G$; the characters of $U$ of degree $q^3/2$ play here a major role. We then determine the $\\ell$-decomposition matrix of $G$ in the case $\\ell \\mid q+1$, when $\\ell \\ge 5$ and $(q+1)_\\ell>5$, up to two non-negative indeterminates in one column."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.05473","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"}