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It turns out that each nilpotent element in $r(H_{n,d})$ can be written as a sum of indecomposable projective representations. The Jacobson radical $J(r(H_{n,d}))$ of $r(H_{n,d})$ is generated by one element, and its rank is $n-n/d$. Moreover, we will present all the finite dimensional indecomposable representations over the complexified Green ring $R(H"},"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":"1210.4245","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-10-16T04:35:36Z","cross_cats_sorted":[],"title_canon_sha256":"9ea3f79cdeaf363c7e46b89d9db37b1d3831064ad9da91a3215018bbd9aef12d","abstract_canon_sha256":"06b991bf8e259bc42067eb95f6053d2aa2c0e21c6836eac0966c10435a37b89e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:06.538229Z","signature_b64":"izn+hqfRi8v4hUsrcVnnTQxO7yZxG4+b7owaDStgymdxhe2uxJXA+QatVOKCydjCY5LAcGJw64w8W8l/oVi/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1298d2616882097447853cefce60cc1e99ee909512c1a97a16aa1ea34155de41","last_reissued_at":"2026-05-18T03:43:06.537466Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:06.537466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Green rings of the generalized Taft Hopf algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Libin Li, Yinhuo Zhang","submitted_at":"2012-10-16T04:35:36Z","abstract_excerpt":"In this paper, we investigate the Green ring $r(H_{n,d})$ of the generalized Taft algebra $H_{n,d}$, extending the results of Chen, Van Oystaeyen and Zhang in \\cite{Coz}. 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