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Consider the cone $lr(G,\\hat G)$ generated by the pairs $(\\nu,\\hat nu)$ of dominant characters such that $V_\\nu^*$ is a submodule of $V_{\\hat nu}$. It is known that $lr(G,\\hat G)$ is a closed convex polyhedral cone. In this work, we show that every regular face of $lr(G,\\hat G)$ gives rise to a {\\it reduction rule} for multiplicities. More precisely, we prove that for $(\\nu,\\hat nu)$ on such a face, the"},"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":"1102.0196","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-02-01T16:05:19Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"002ce875b16ba02b4c128a016c66a1903aa4b9fc113eb6b00b5a01c49620fa22","abstract_canon_sha256":"fb4b625fb6d606e0079f73ccd3f991340b11ce60b22114e20064298e264fa994"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:32.074031Z","signature_b64":"GKUl2RxyHHGbCXBk7tPdTOp8sbv20hSPjLHJlvSvb8x6a7VorSm2IDiDrY4BJw15vmLhojLBWGj0wrlrhtOpBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50b414b90667ced73e22bc42cc0bafb4c05ba57d5066e9c42d6c86168b171207","last_reissued_at":"2026-05-18T03:45:32.073281Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:32.073281Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reductions for branching coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.AG","authors_text":"Nicolas Ressayre (I3M)","submitted_at":"2011-02-01T16:05:19Z","abstract_excerpt":"Let $G$ be a connected reductive subgroup of a complex connected reductive group $\\hat{G}$. We are interested in the branching problem. Fix maximal tori and Borel subgroups of $G$ and $\\hat G$. Consider the cone $lr(G,\\hat G)$ generated by the pairs $(\\nu,\\hat nu)$ of dominant characters such that $V_\\nu^*$ is a submodule of $V_{\\hat nu}$. It is known that $lr(G,\\hat G)$ is a closed convex polyhedral cone. In this work, we show that every regular face of $lr(G,\\hat G)$ gives rise to a {\\it reduction rule} for multiplicities. 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