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As a Virasoro module, we have $$\\mathcal {W}(p)=\\bigoplus_{n=0} ^{\\infty}(2n+1) L(c_{p}, n^{2}p+np-n).$$ It was pointed out in \\cite{am1} that $\\mathcal {W}(p)$ admits an action of $\\mathfrak{sl}_{2}$. In this paper we give a combinatorics description of $\\mathcal {W}(p)$, from which the action of $\\mathfrak{sl}_{2}$ follows quite directly. In the end of this paper we give similar descriptions of the invariant subalgebra $\\mathcal {W}(p)^{\\Gamma}$, these will be useful for the charac"},"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":"1311.2113","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-11-09T00:47:34Z","cross_cats_sorted":["math-ph","math.MP","math.QA"],"title_canon_sha256":"43c5cdf055753a62d5505c071caefe5b633760cccb73892485624c54001bf37a","abstract_canon_sha256":"268e2d6a6a18b95745d090edffcc95b3d76ff259c35c5aa8c5970a1e4b924c14"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:42.553309Z","signature_b64":"NhtzmHelqWQ52Y7dAfYX+T9pIbnyWut4Wx7wkd1PZB3RGMvnkWRMUAa4SoclsYQkOTbf/5wBvL8yusXPOXbiDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0cd045e4a620c052bdd39f749fe1afb591589f5babbd135f2d9627e5f715ee83","last_reissued_at":"2026-05-18T03:06:42.552674Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:42.552674Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A new proof of the $\\mathfrak{sl}_{2}$ action on the triplet vertex algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.QA"],"primary_cat":"math.RT","authors_text":"Xianzu Lin","submitted_at":"2013-11-09T00:47:34Z","abstract_excerpt":"Let $\\mathcal {W}(p)$ be the triplet vertex algebra of central charge $c_{p}=1-\\frac{6(p-1)^{2}}{p}$, $p\\geq2$. 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