{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:MGJSAER2WJNYCYSM63GPNACRDG","short_pith_number":"pith:MGJSAER2","canonical_record":{"source":{"id":"1609.09538","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-09-29T21:45:21Z","cross_cats_sorted":[],"title_canon_sha256":"15ea90efca69ca0ccd07b9510f819551ea09fde5e3c235f8ca14d3e41322ac54","abstract_canon_sha256":"a60adeef82656ca8f7523d772dfdb92f5eb48dca4f83dd9bca5f325c57be83df"},"schema_version":"1.0"},"canonical_sha256":"619320123ab25b81624cf6ccf680511981b22d9470dcb03c9b66b352eb95e1f1","source":{"kind":"arxiv","id":"1609.09538","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.09538","created_at":"2026-05-18T00:32:31Z"},{"alias_kind":"arxiv_version","alias_value":"1609.09538v2","created_at":"2026-05-18T00:32:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.09538","created_at":"2026-05-18T00:32:31Z"},{"alias_kind":"pith_short_12","alias_value":"MGJSAER2WJNY","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MGJSAER2WJNYCYSM","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MGJSAER2","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:MGJSAER2WJNYCYSM63GPNACRDG","target":"record","payload":{"canonical_record":{"source":{"id":"1609.09538","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-09-29T21:45:21Z","cross_cats_sorted":[],"title_canon_sha256":"15ea90efca69ca0ccd07b9510f819551ea09fde5e3c235f8ca14d3e41322ac54","abstract_canon_sha256":"a60adeef82656ca8f7523d772dfdb92f5eb48dca4f83dd9bca5f325c57be83df"},"schema_version":"1.0"},"canonical_sha256":"619320123ab25b81624cf6ccf680511981b22d9470dcb03c9b66b352eb95e1f1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:31.302676Z","signature_b64":"DueKhA7RmIcPyaK9xrO5onx/hMXdwy5sUzzJcmNQcTJK6H34kN1joBa/WlUuhp1AU5uKczvF9aaPHW2Mzo2tBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"619320123ab25b81624cf6ccf680511981b22d9470dcb03c9b66b352eb95e1f1","last_reissued_at":"2026-05-18T00:32:31.302030Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:31.302030Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.09538","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:32:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MbfYAcjl6IkU3M0lcpjS0tFfBmXmkZiTtvlubTNvzJABnc/ZXfwrCQyxFK+hhgzQXbhlj11lEvHWuNjibkrRAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T03:06:32.328259Z"},"content_sha256":"c86f6134648ffd791fe4f1acd892681a0a09a95d659d0228ef3d1f2da1c5aca3","schema_version":"1.0","event_id":"sha256:c86f6134648ffd791fe4f1acd892681a0a09a95d659d0228ef3d1f2da1c5aca3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:MGJSAER2WJNYCYSM63GPNACRDG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of their Coordinate Rings, and Sphericity Consequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Reuven Hodges, Venkatramani Lakshmibai","submitted_at":"2016-09-29T21:45:21Z","abstract_excerpt":"Let $L_w$ be the Levi part of the stabilizer $Q_w$ in $GL_N$ (for left multiplication) of a Schubert variety $X(w)$ in the Grassmannian $G_{d,N}$. For the natural action of $L_w$ on $\\mathbb{C}[X(w)]$, the homogeneous coordinate ring of $X(w)$ (for the Pl\\\"ucker embedding), we give a combinatorial description of the decomposition of $\\mathbb{C}[X(w)]$ into irreducible $L_w$-modules; in fact, our description holds more generally for the action of the Levi part $L$ of any parabolic subgroup $Q$ that is contained in $Q_w$. This decomposition is then used to show that all smooth Schubert varieties"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.09538","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:32:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WDU6QQNogkDZ+zAVzL4ESRSQl5P0njhYv/O6HE7ecHMZOBgOilmwCLw2WSiCVrfb37GNJJnRSUwnjQSrPFltBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T03:06:32.328631Z"},"content_sha256":"b1d9b2d8714d15be41e67beaaaea3e4c3770c3e6dcb2fddf7296a88cd1c8e4a6","schema_version":"1.0","event_id":"sha256:b1d9b2d8714d15be41e67beaaaea3e4c3770c3e6dcb2fddf7296a88cd1c8e4a6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MGJSAER2WJNYCYSM63GPNACRDG/bundle.json","state_url":"https://pith.science/pith/MGJSAER2WJNYCYSM63GPNACRDG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MGJSAER2WJNYCYSM63GPNACRDG/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T03:06:32Z","links":{"resolver":"https://pith.science/pith/MGJSAER2WJNYCYSM63GPNACRDG","bundle":"https://pith.science/pith/MGJSAER2WJNYCYSM63GPNACRDG/bundle.json","state":"https://pith.science/pith/MGJSAER2WJNYCYSM63GPNACRDG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MGJSAER2WJNYCYSM63GPNACRDG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:MGJSAER2WJNYCYSM63GPNACRDG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a60adeef82656ca8f7523d772dfdb92f5eb48dca4f83dd9bca5f325c57be83df","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-09-29T21:45:21Z","title_canon_sha256":"15ea90efca69ca0ccd07b9510f819551ea09fde5e3c235f8ca14d3e41322ac54"},"schema_version":"1.0","source":{"id":"1609.09538","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.09538","created_at":"2026-05-18T00:32:31Z"},{"alias_kind":"arxiv_version","alias_value":"1609.09538v2","created_at":"2026-05-18T00:32:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.09538","created_at":"2026-05-18T00:32:31Z"},{"alias_kind":"pith_short_12","alias_value":"MGJSAER2WJNY","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MGJSAER2WJNYCYSM","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MGJSAER2","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:b1d9b2d8714d15be41e67beaaaea3e4c3770c3e6dcb2fddf7296a88cd1c8e4a6","target":"graph","created_at":"2026-05-18T00:32:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $L_w$ be the Levi part of the stabilizer $Q_w$ in $GL_N$ (for left multiplication) of a Schubert variety $X(w)$ in the Grassmannian $G_{d,N}$. For the natural action of $L_w$ on $\\mathbb{C}[X(w)]$, the homogeneous coordinate ring of $X(w)$ (for the Pl\\\"ucker embedding), we give a combinatorial description of the decomposition of $\\mathbb{C}[X(w)]$ into irreducible $L_w$-modules; in fact, our description holds more generally for the action of the Levi part $L$ of any parabolic subgroup $Q$ that is contained in $Q_w$. This decomposition is then used to show that all smooth Schubert varieties","authors_text":"Reuven Hodges, Venkatramani Lakshmibai","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-09-29T21:45:21Z","title":"Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of their Coordinate Rings, and Sphericity Consequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.09538","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c86f6134648ffd791fe4f1acd892681a0a09a95d659d0228ef3d1f2da1c5aca3","target":"record","created_at":"2026-05-18T00:32:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a60adeef82656ca8f7523d772dfdb92f5eb48dca4f83dd9bca5f325c57be83df","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-09-29T21:45:21Z","title_canon_sha256":"15ea90efca69ca0ccd07b9510f819551ea09fde5e3c235f8ca14d3e41322ac54"},"schema_version":"1.0","source":{"id":"1609.09538","kind":"arxiv","version":2}},"canonical_sha256":"619320123ab25b81624cf6ccf680511981b22d9470dcb03c9b66b352eb95e1f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"619320123ab25b81624cf6ccf680511981b22d9470dcb03c9b66b352eb95e1f1","first_computed_at":"2026-05-18T00:32:31.302030Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:31.302030Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DueKhA7RmIcPyaK9xrO5onx/hMXdwy5sUzzJcmNQcTJK6H34kN1joBa/WlUuhp1AU5uKczvF9aaPHW2Mzo2tBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:31.302676Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.09538","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c86f6134648ffd791fe4f1acd892681a0a09a95d659d0228ef3d1f2da1c5aca3","sha256:b1d9b2d8714d15be41e67beaaaea3e4c3770c3e6dcb2fddf7296a88cd1c8e4a6"],"state_sha256":"80197dcfd38b6b526cfa9410a9bf942f720a79ab92cdff3ff399d31cbe18bb09"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kkCbP0YulP6Yp3NeRQ7AZE5ebH/jkaWvLouztnCb0VDmeHKR3cChDlQHAOQ+t/9ZIW0qze0tGrtSanur+lxlBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T03:06:32.330555Z","bundle_sha256":"b0870f48f0df09816140810bf0a9a473fe39a4a6c1f53f0228de1d8f2299a91e"}}