{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:WX7X6YWFDQX2WXB4Z7YD5TG2ZN","short_pith_number":"pith:WX7X6YWF","canonical_record":{"source":{"id":"1703.05898","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.RT","submitted_at":"2017-03-17T05:37:36Z","cross_cats_sorted":[],"title_canon_sha256":"a9b2b994242a401b807005e5a9a955ce06f528c1aa780bc3235b84f8844eb1bb","abstract_canon_sha256":"ca2979028ff56940fb8762e1ea6616093411e00c9b4c3592626225c407f6004d"},"schema_version":"1.0"},"canonical_sha256":"b5ff7f62c51c2fab5c3ccff03eccdacb6832a4fe207ca8a4fbff01aba1d707b5","source":{"kind":"arxiv","id":"1703.05898","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.05898","created_at":"2026-05-18T00:22:52Z"},{"alias_kind":"arxiv_version","alias_value":"1703.05898v3","created_at":"2026-05-18T00:22:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.05898","created_at":"2026-05-18T00:22:52Z"},{"alias_kind":"pith_short_12","alias_value":"WX7X6YWFDQX2","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"WX7X6YWFDQX2WXB4","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"WX7X6YWF","created_at":"2026-05-18T12:31:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:WX7X6YWFDQX2WXB4Z7YD5TG2ZN","target":"record","payload":{"canonical_record":{"source":{"id":"1703.05898","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.RT","submitted_at":"2017-03-17T05:37:36Z","cross_cats_sorted":[],"title_canon_sha256":"a9b2b994242a401b807005e5a9a955ce06f528c1aa780bc3235b84f8844eb1bb","abstract_canon_sha256":"ca2979028ff56940fb8762e1ea6616093411e00c9b4c3592626225c407f6004d"},"schema_version":"1.0"},"canonical_sha256":"b5ff7f62c51c2fab5c3ccff03eccdacb6832a4fe207ca8a4fbff01aba1d707b5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:52.866490Z","signature_b64":"uwWJd/dD6z1ZiO24E0F8uhsxutmU9JxwBeN8/+C0y4MluLyFbo6rzL0xMCffzpU9+GPIG+msZNfvDMK5Ow0xBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5ff7f62c51c2fab5c3ccff03eccdacb6832a4fe207ca8a4fbff01aba1d707b5","last_reissued_at":"2026-05-18T00:22:52.866098Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:52.866098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1703.05898","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hrhEKwdRn/z7tLGH1LuPLFaliFYE3gq659v9YVtr+kC5IHeykt6DgDizi+vb74WB5AqptrrWa/7SUy9jhhLzAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T21:23:17.953188Z"},"content_sha256":"98bcf5aad5cd8550f6f131ebeddabb46c5d711ff40a930dc969831edf999b7b3","schema_version":"1.0","event_id":"sha256:98bcf5aad5cd8550f6f131ebeddabb46c5d711ff40a930dc969831edf999b7b3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:WX7X6YWFDQX2WXB4Z7YD5TG2ZN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Billiards and Tilting Characters for ${\\rm SL}_3$","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Geordie Williamson, George Lusztig","submitted_at":"2017-03-17T05:37:36Z","abstract_excerpt":"We formulate a conjecture for the second generation characters of indecomposable tilting modules for ${\\rm SL}_3$. This gives many new conjectural decomposition numbers for symmetric groups. Our conjecture can be interpreted as saying that these characters are governed by a discrete dynamical system (\"billiards bouncing in alcoves\"). The conjecture implies that decomposition numbers for symmetric groups display (at least) exponential growth."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05898","kind":"arxiv","version":3},"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":"2026-05-18T00:22:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iJhxeXGSrxb9+ze9zAsilpjPBukb8HaXR/GXESIXDhZmw6D1ELdYJC4vZc0Y1xmcRROy4w2qzgzxS2g1iAzdCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T21:23:17.953557Z"},"content_sha256":"fd58829008c1f61e51442d313eb062a984d5e6301df5dac786dd95e0c69b745e","schema_version":"1.0","event_id":"sha256:fd58829008c1f61e51442d313eb062a984d5e6301df5dac786dd95e0c69b745e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WX7X6YWFDQX2WXB4Z7YD5TG2ZN/bundle.json","state_url":"https://pith.science/pith/WX7X6YWFDQX2WXB4Z7YD5TG2ZN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WX7X6YWFDQX2WXB4Z7YD5TG2ZN/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":"2026-06-07T21:23:17Z","links":{"resolver":"https://pith.science/pith/WX7X6YWFDQX2WXB4Z7YD5TG2ZN","bundle":"https://pith.science/pith/WX7X6YWFDQX2WXB4Z7YD5TG2ZN/bundle.json","state":"https://pith.science/pith/WX7X6YWFDQX2WXB4Z7YD5TG2ZN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WX7X6YWFDQX2WXB4Z7YD5TG2ZN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:WX7X6YWFDQX2WXB4Z7YD5TG2ZN","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":"ca2979028ff56940fb8762e1ea6616093411e00c9b4c3592626225c407f6004d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.RT","submitted_at":"2017-03-17T05:37:36Z","title_canon_sha256":"a9b2b994242a401b807005e5a9a955ce06f528c1aa780bc3235b84f8844eb1bb"},"schema_version":"1.0","source":{"id":"1703.05898","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.05898","created_at":"2026-05-18T00:22:52Z"},{"alias_kind":"arxiv_version","alias_value":"1703.05898v3","created_at":"2026-05-18T00:22:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.05898","created_at":"2026-05-18T00:22:52Z"},{"alias_kind":"pith_short_12","alias_value":"WX7X6YWFDQX2","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"WX7X6YWFDQX2WXB4","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"WX7X6YWF","created_at":"2026-05-18T12:31:53Z"}],"graph_snapshots":[{"event_id":"sha256:fd58829008c1f61e51442d313eb062a984d5e6301df5dac786dd95e0c69b745e","target":"graph","created_at":"2026-05-18T00:22:52Z","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":"We formulate a conjecture for the second generation characters of indecomposable tilting modules for ${\\rm SL}_3$. This gives many new conjectural decomposition numbers for symmetric groups. Our conjecture can be interpreted as saying that these characters are governed by a discrete dynamical system (\"billiards bouncing in alcoves\"). The conjecture implies that decomposition numbers for symmetric groups display (at least) exponential growth.","authors_text":"Geordie Williamson, George Lusztig","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.RT","submitted_at":"2017-03-17T05:37:36Z","title":"Billiards and Tilting Characters for ${\\rm SL}_3$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05898","kind":"arxiv","version":3},"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":"sha256:98bcf5aad5cd8550f6f131ebeddabb46c5d711ff40a930dc969831edf999b7b3","target":"record","created_at":"2026-05-18T00:22:52Z","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":"ca2979028ff56940fb8762e1ea6616093411e00c9b4c3592626225c407f6004d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.RT","submitted_at":"2017-03-17T05:37:36Z","title_canon_sha256":"a9b2b994242a401b807005e5a9a955ce06f528c1aa780bc3235b84f8844eb1bb"},"schema_version":"1.0","source":{"id":"1703.05898","kind":"arxiv","version":3}},"canonical_sha256":"b5ff7f62c51c2fab5c3ccff03eccdacb6832a4fe207ca8a4fbff01aba1d707b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b5ff7f62c51c2fab5c3ccff03eccdacb6832a4fe207ca8a4fbff01aba1d707b5","first_computed_at":"2026-05-18T00:22:52.866098Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:52.866098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uwWJd/dD6z1ZiO24E0F8uhsxutmU9JxwBeN8/+C0y4MluLyFbo6rzL0xMCffzpU9+GPIG+msZNfvDMK5Ow0xBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:52.866490Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.05898","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:98bcf5aad5cd8550f6f131ebeddabb46c5d711ff40a930dc969831edf999b7b3","sha256:fd58829008c1f61e51442d313eb062a984d5e6301df5dac786dd95e0c69b745e"],"state_sha256":"277eca8f5a8bbdff490df13740502c7a03089de94fa3c91e15f949b29264688e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FPR1UqGxQAupvYcqo5FrITbJ4GlO/ZXr4kGKGgOpRa4eb5GYxEremDhHJGZfHUc8aWfTYWXL5VfJS5dnbbSfDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T21:23:17.956158Z","bundle_sha256":"5203a48ecc2695927e2a488465a73b9ad7d44e0a1662a04943f3719c304e2236"}}