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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2405.16015","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2024-05-25T02:40:53Z","cross_cats_sorted":[],"title_canon_sha256":"c1c90a1c0296f6b8cfa456fa648e47666f67c18f25e11af330a1c831e1a7b8d0","abstract_canon_sha256":"0bb61654602ce9e46f14804b2ba519aecde071d4e3fb6aeb4c9a2576ead647ed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:23:00.603598Z","signature_b64":"BJWtcviIfrK4y2Av+8zw+oKik3UqLtxtXHfiiWMcU3U5numd92whUICOGBfaVDo5dtd+2kX0xTU01CaJ5X3DAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d1973b8e31e5a231447486f24f229bb1a9612f4f2f11acd893675dbd9e19e94d","last_reissued_at":"2026-07-05T08:23:00.603144Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:23:00.603144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2405.16015","created_at":"2026-07-05T08:23:00.603206+00:00"},{"alias_kind":"arxiv_version","alias_value":"2405.16015v1","created_at":"2026-07-05T08:23:00.603206+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.16015","created_at":"2026-07-05T08:23:00.603206+00:00"},{"alias_kind":"pith_short_12","alias_value":"2GLTXDRR4WRD","created_at":"2026-07-05T08:23:00.603206+00:00"},{"alias_kind":"pith_short_16","alias_value":"2GLTXDRR4WRDCRDU","created_at":"2026-07-05T08:23:00.603206+00:00"},{"alias_kind":"pith_short_8","alias_value":"2GLTXDRR","created_at":"2026-07-05T08:23:00.603206+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2502.02849","citing_title":"Growth Problems for Representations of Finite Monoids","ref_index":10,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2GLTXDRR4WRDCRDUQ3ZE6IU3WG","json":"https://pith.science/pith/2GLTXDRR4WRDCRDUQ3ZE6IU3WG.json","graph_json":"https://pith.science/api/pith-number/2GLTXDRR4WRDCRDUQ3ZE6IU3WG/graph.json","events_json":"https://pith.science/api/pith-number/2GLTXDRR4WRDCRDUQ3ZE6IU3WG/events.json","paper":"https://pith.science/paper/2GLTXDRR"},"agent_actions":{"view_html":"https://pith.science/pith/2GLTXDRR4WRDCRDUQ3ZE6IU3WG","download_json":"https://pith.science/pith/2GLTXDRR4WRDCRDUQ3ZE6IU3WG.json","view_paper":"https://pith.science/paper/2GLTXDRR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2405.16015&json=true","fetch_graph":"https://pith.science/api/pith-number/2GLTXDRR4WRDCRDUQ3ZE6IU3WG/graph.json","fetch_events":"https://pith.science/api/pith-number/2GLTXDRR4WRDCRDUQ3ZE6IU3WG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2GLTXDRR4WRDCRDUQ3ZE6IU3WG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2GLTXDRR4WRDCRDUQ3ZE6IU3WG/action/storage_attestation","attest_author":"https://pith.science/pith/2GLTXDRR4WRDCRDUQ3ZE6IU3WG/action/author_attestation","sign_citation":"https://pith.science/pith/2GLTXDRR4WRDCRDUQ3ZE6IU3WG/action/citation_signature","submit_replication":"https://pith.science/pith/2GLTXDRR4WRDCRDUQ3ZE6IU3WG/action/replication_record"}},"created_at":"2026-07-05T08:23:00.603206+00:00","updated_at":"2026-07-05T08:23:00.603206+00:00"}