{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:24LTWSWO5GHXJ3BRXDX32EBLTB","short_pith_number":"pith:24LTWSWO","canonical_record":{"source":{"id":"1908.08405","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-22T14:26:29Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"19d6d075654304807b1a13dfae12e63145b70917450a423740e5bdd5f5250e31","abstract_canon_sha256":"ab9b63860f7782482f83dbd23c01244ffba81ab149bb1856c8c23f89682044ce"},"schema_version":"1.0"},"canonical_sha256":"d7173b4acee98f74ec31b8efbd102b98508527b49264616b93dce0f6137cc4ee","source":{"kind":"arxiv","id":"1908.08405","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.08405","created_at":"2026-07-05T08:21:02Z"},{"alias_kind":"arxiv_version","alias_value":"1908.08405v1","created_at":"2026-07-05T08:21:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08405","created_at":"2026-07-05T08:21:02Z"},{"alias_kind":"pith_short_12","alias_value":"24LTWSWO5GHX","created_at":"2026-07-05T08:21:02Z"},{"alias_kind":"pith_short_16","alias_value":"24LTWSWO5GHXJ3BR","created_at":"2026-07-05T08:21:02Z"},{"alias_kind":"pith_short_8","alias_value":"24LTWSWO","created_at":"2026-07-05T08:21:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:24LTWSWO5GHXJ3BRXDX32EBLTB","target":"record","payload":{"canonical_record":{"source":{"id":"1908.08405","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-22T14:26:29Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"19d6d075654304807b1a13dfae12e63145b70917450a423740e5bdd5f5250e31","abstract_canon_sha256":"ab9b63860f7782482f83dbd23c01244ffba81ab149bb1856c8c23f89682044ce"},"schema_version":"1.0"},"canonical_sha256":"d7173b4acee98f74ec31b8efbd102b98508527b49264616b93dce0f6137cc4ee","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:21:02.013100Z","signature_b64":"T2HNAtRty8Bd2y6GUFy3n3PsyFtqS9DTRXnE8evGkbSb89P9xQ2K6zNtDgG0LSmR4lBA7R0zbtVYv9RrORlsDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7173b4acee98f74ec31b8efbd102b98508527b49264616b93dce0f6137cc4ee","last_reissued_at":"2026-07-05T08:21:02.012627Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:21:02.012627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.08405","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:21:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EYfL3He1rBsiN9JUoEr9vrsvhOQZ0aHYJ0LGloTjD8RkpGs719Ym4uX+VRAnU5JV8kk8/71Vw6Zoz0T5mLqvBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T08:42:31.049130Z"},"content_sha256":"73e8264f4e2b9859ed63ea58eacae36c09cc50aa1533b5d8e7e704adeab0f434","schema_version":"1.0","event_id":"sha256:73e8264f4e2b9859ed63ea58eacae36c09cc50aa1533b5d8e7e704adeab0f434"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:24LTWSWO5GHXJ3BRXDX32EBLTB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Eduardo Torres Davila, Fabrice O. Ulysse, Gordon Rojas Kirby, Joseph Rennie, Juan Ramirez, Marissa Loving, Pamela E. Harris","submitted_at":"2019-08-22T14:26:29Z","abstract_excerpt":"The multiplicity of a weight in a finite-dimensional irreducible representation of a simple Lie algebra g can be computed via Kostant's weight multiplicity formula. This formula consists of an alternating sum over the Weyl group (a finite group) and involves a partition function known as Kostant's partition function. Motivated by the observation that, in practice, most terms in the sum are zero, our main results describe the elements of the Weyl alternation sets. The Weyl alternation sets are subsets of the Weyl group which contributes nontrivially to the multiplicity of a weight in a highest "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08405","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.08405/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T08:21:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YwQlnAXoSel0e83XnLGJmhAiDec2FUHOJpZNEJXCgtFf/dA2lx018rhiCUDskHE14kG/jLslyvub4XJpAr2bBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T08:42:31.049838Z"},"content_sha256":"df42cc78dfbdc3f6ef4763bf03b2c8d040dc0f66834223434bb28dde2bf2ff92","schema_version":"1.0","event_id":"sha256:df42cc78dfbdc3f6ef4763bf03b2c8d040dc0f66834223434bb28dde2bf2ff92"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/24LTWSWO5GHXJ3BRXDX32EBLTB/bundle.json","state_url":"https://pith.science/pith/24LTWSWO5GHXJ3BRXDX32EBLTB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/24LTWSWO5GHXJ3BRXDX32EBLTB/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":"2026-08-16T08:42:31Z","links":{"resolver":"https://pith.science/pith/24LTWSWO5GHXJ3BRXDX32EBLTB","bundle":"https://pith.science/pith/24LTWSWO5GHXJ3BRXDX32EBLTB/bundle.json","state":"https://pith.science/pith/24LTWSWO5GHXJ3BRXDX32EBLTB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/24LTWSWO5GHXJ3BRXDX32EBLTB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:24LTWSWO5GHXJ3BRXDX32EBLTB","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":"ab9b63860f7782482f83dbd23c01244ffba81ab149bb1856c8c23f89682044ce","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-22T14:26:29Z","title_canon_sha256":"19d6d075654304807b1a13dfae12e63145b70917450a423740e5bdd5f5250e31"},"schema_version":"1.0","source":{"id":"1908.08405","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.08405","created_at":"2026-07-05T08:21:02Z"},{"alias_kind":"arxiv_version","alias_value":"1908.08405v1","created_at":"2026-07-05T08:21:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08405","created_at":"2026-07-05T08:21:02Z"},{"alias_kind":"pith_short_12","alias_value":"24LTWSWO5GHX","created_at":"2026-07-05T08:21:02Z"},{"alias_kind":"pith_short_16","alias_value":"24LTWSWO5GHXJ3BR","created_at":"2026-07-05T08:21:02Z"},{"alias_kind":"pith_short_8","alias_value":"24LTWSWO","created_at":"2026-07-05T08:21:02Z"}],"graph_snapshots":[{"event_id":"sha256:df42cc78dfbdc3f6ef4763bf03b2c8d040dc0f66834223434bb28dde2bf2ff92","target":"graph","created_at":"2026-07-05T08:21:02Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.08405/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The multiplicity of a weight in a finite-dimensional irreducible representation of a simple Lie algebra g can be computed via Kostant's weight multiplicity formula. This formula consists of an alternating sum over the Weyl group (a finite group) and involves a partition function known as Kostant's partition function. Motivated by the observation that, in practice, most terms in the sum are zero, our main results describe the elements of the Weyl alternation sets. The Weyl alternation sets are subsets of the Weyl group which contributes nontrivially to the multiplicity of a weight in a highest ","authors_text":"Eduardo Torres Davila, Fabrice O. Ulysse, Gordon Rojas Kirby, Joseph Rennie, Juan Ramirez, Marissa Loving, Pamela E. Harris","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-22T14:26:29Z","title":"Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08405","kind":"arxiv","version":1},"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":"sha256:73e8264f4e2b9859ed63ea58eacae36c09cc50aa1533b5d8e7e704adeab0f434","target":"record","created_at":"2026-07-05T08:21:02Z","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":"ab9b63860f7782482f83dbd23c01244ffba81ab149bb1856c8c23f89682044ce","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-22T14:26:29Z","title_canon_sha256":"19d6d075654304807b1a13dfae12e63145b70917450a423740e5bdd5f5250e31"},"schema_version":"1.0","source":{"id":"1908.08405","kind":"arxiv","version":1}},"canonical_sha256":"d7173b4acee98f74ec31b8efbd102b98508527b49264616b93dce0f6137cc4ee","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d7173b4acee98f74ec31b8efbd102b98508527b49264616b93dce0f6137cc4ee","first_computed_at":"2026-07-05T08:21:02.012627Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:21:02.012627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"T2HNAtRty8Bd2y6GUFy3n3PsyFtqS9DTRXnE8evGkbSb89P9xQ2K6zNtDgG0LSmR4lBA7R0zbtVYv9RrORlsDw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:21:02.013100Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.08405","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73e8264f4e2b9859ed63ea58eacae36c09cc50aa1533b5d8e7e704adeab0f434","sha256:df42cc78dfbdc3f6ef4763bf03b2c8d040dc0f66834223434bb28dde2bf2ff92"],"state_sha256":"91220fd4085b61ea6e6f8eddd7b484e1307436842a6cd71133dd911d098d05eb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ps7XFSNWcjZT3q4GTPyaLDqwWMmE2CoOVw6m/nv2wrcsRzYct7/y1SBe7176EDFupSJY8NbTZb8egXRMaQUCDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T08:42:31.055244Z","bundle_sha256":"e1782f7d85a328e927185266871a9a5d7131b9f42a3b85b8266c0e48d6963552"}}