{"paper":{"title":"Bounds for $\\mathrm{SL}_2$-indecomposables in tensor powers of the natural representation in characteristic $2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Michael J. Larsen","submitted_at":"2024-05-25T02:40:53Z","abstract_excerpt":"Let $K$ be an algebraically closed field of characteristic $2$, $G$ be the algebraic group $\\mathrm{SL}_2$ over $K$, and $V$ be the natural representation of $G$. Let $b_k^{G,V}$ denote the number of $G$-indecomposable factors of $V^{\\otimes k}$, counted with multiplicity, and let $\\delta = \\frac 32 - \\frac{\\log 3}{2\\log 2}$. Then there exists a smooth multiplicatively periodic function $\\omega(x)$ such that $b_{2k}^{G,V} = b_{2k+1}^{G,V}$ is asymptotic to $\\omega(k) k^{-\\delta}4^k$. We also prove a lower bound of the form $c_W k^{-\\delta}(\\dim W)^k$ for $b_k^{G,W} $ for any tilting representa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.16015","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/2405.16015/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"}