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We show that, when $r$ is not a power of $p$, the Lie power $L^r(V)$ has a direct summand $B^r(V)$ which is a direct summand of the tensor power $V^{\\otimes r}$ and which satisfies $\\dim B^r(V)/\\dim L^r(V) \\to 1$ as $r \\to \\infty$. Similarly, for the same values of $r$, we obtain a projective submodule $C(r)$ of the Lie module $\\Lie(r)$ over $F$ such that $\\dim C(r)/\\dim \\Lie(r) \\to 1$ as $r \\to \\infty$."},"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":"1009.0974","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-09-06T06:23:51Z","cross_cats_sorted":[],"title_canon_sha256":"2a93c9b3e005433beb15a136ceb93ebc8112ced13644671ac7ea5be2dd870df1","abstract_canon_sha256":"91b7982f4b901902a168a7621bd71565977456f2470d19cc5b887a1caa26cabe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:09.883895Z","signature_b64":"d/6YWXk0dU684BD+8JIYrFxCUheQ5ZWA1yDK6UT/D8WsFAxyD+E8Y5b+vZT4/rZBLaMyWrdLHBJEBqZkK+tfDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d3547b33e38b42df4fb22d6ad71f48047abcf7327960cdb4baee64f7c244dc4","last_reissued_at":"2026-05-18T04:34:09.883491Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:09.883491Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic behaviour of Lie powers and Lie modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Kai Meng Tan, Kay Jin Lim, Roger M. Bryant","submitted_at":"2010-09-06T06:23:51Z","abstract_excerpt":"Let $V$ be a finite-dimensional $FG$-module, where $F$ is a field of prime characteristic $p$ and $G$ is a group. We show that, when $r$ is not a power of $p$, the Lie power $L^r(V)$ has a direct summand $B^r(V)$ which is a direct summand of the tensor power $V^{\\otimes r}$ and which satisfies $\\dim B^r(V)/\\dim L^r(V) \\to 1$ as $r \\to \\infty$. Similarly, for the same values of $r$, we obtain a projective submodule $C(r)$ of the Lie module $\\Lie(r)$ over $F$ such that $\\dim C(r)/\\dim \\Lie(r) \\to 1$ as $r \\to \\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0974","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1009.0974","created_at":"2026-05-18T04:34:09.883557+00:00"},{"alias_kind":"arxiv_version","alias_value":"1009.0974v2","created_at":"2026-05-18T04:34:09.883557+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.0974","created_at":"2026-05-18T04:34:09.883557+00:00"},{"alias_kind":"pith_short_12","alias_value":"LU2UPMZ6HC2C","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_16","alias_value":"LU2UPMZ6HC2C35H3","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_8","alias_value":"LU2UPMZ6","created_at":"2026-05-18T12:26:10.704358+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LU2UPMZ6HC2C35H3ELLK24PUQB","json":"https://pith.science/pith/LU2UPMZ6HC2C35H3ELLK24PUQB.json","graph_json":"https://pith.science/api/pith-number/LU2UPMZ6HC2C35H3ELLK24PUQB/graph.json","events_json":"https://pith.science/api/pith-number/LU2UPMZ6HC2C35H3ELLK24PUQB/events.json","paper":"https://pith.science/paper/LU2UPMZ6"},"agent_actions":{"view_html":"https://pith.science/pith/LU2UPMZ6HC2C35H3ELLK24PUQB","download_json":"https://pith.science/pith/LU2UPMZ6HC2C35H3ELLK24PUQB.json","view_paper":"https://pith.science/paper/LU2UPMZ6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1009.0974&json=true","fetch_graph":"https://pith.science/api/pith-number/LU2UPMZ6HC2C35H3ELLK24PUQB/graph.json","fetch_events":"https://pith.science/api/pith-number/LU2UPMZ6HC2C35H3ELLK24PUQB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LU2UPMZ6HC2C35H3ELLK24PUQB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LU2UPMZ6HC2C35H3ELLK24PUQB/action/storage_attestation","attest_author":"https://pith.science/pith/LU2UPMZ6HC2C35H3ELLK24PUQB/action/author_attestation","sign_citation":"https://pith.science/pith/LU2UPMZ6HC2C35H3ELLK24PUQB/action/citation_signature","submit_replication":"https://pith.science/pith/LU2UPMZ6HC2C35H3ELLK24PUQB/action/replication_record"}},"created_at":"2026-05-18T04:34:09.883557+00:00","updated_at":"2026-05-18T04:34:09.883557+00:00"}