{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:T2S2CKAK6KM6Y2J2PDYFXMUUCD","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":"4078cc14aeb24298810521c16fcadcf369ff9028a5d12f11dbaecdcfccf307dc","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-11T23:29:47Z","title_canon_sha256":"2404b7199efdc2f5caf0f47f7dd0b1981481f2632f773021c37f524d6bdc9d15"},"schema_version":"1.0","source":{"id":"1204.2591","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.2591","created_at":"2026-05-18T03:42:11Z"},{"alias_kind":"arxiv_version","alias_value":"1204.2591v4","created_at":"2026-05-18T03:42:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.2591","created_at":"2026-05-18T03:42:11Z"},{"alias_kind":"pith_short_12","alias_value":"T2S2CKAK6KM6","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"T2S2CKAK6KM6Y2J2","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"T2S2CKAK","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:be9aa7f5d941a18b8c4c509128ad01283c9524f1027efc3f2765b08530f368ed","target":"graph","created_at":"2026-05-18T03:42:11Z","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 study the unique maximal decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements, providing a new perspective on work of Thomas Lam. This decomposition is closely related to the affine code, which generalizes the $k$-bounded partition associated to Grassmannian elements. We also show that the affine code readily encodes a number of basic combinatorial properties of an affine permutation. As an application, we prove a new special case of the Littlewood-Richardson Rule for $k$-Schur functions, using the canonical decomposition to control for which per","authors_text":"Tom Denton","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-11T23:29:47Z","title":"Canonical Decompositions of Affine Permutations, Affine Codes, and Split $k$-Schur Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.2591","kind":"arxiv","version":4},"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:a3727f752a2ec4957f694a7203ce5f345720d2f2b04a63116637d98b4a84b77b","target":"record","created_at":"2026-05-18T03:42:11Z","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":"4078cc14aeb24298810521c16fcadcf369ff9028a5d12f11dbaecdcfccf307dc","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-11T23:29:47Z","title_canon_sha256":"2404b7199efdc2f5caf0f47f7dd0b1981481f2632f773021c37f524d6bdc9d15"},"schema_version":"1.0","source":{"id":"1204.2591","kind":"arxiv","version":4}},"canonical_sha256":"9ea5a1280af299ec693a78f05bb29410c5a2d6e86481ce2b443011e5a86125cd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9ea5a1280af299ec693a78f05bb29410c5a2d6e86481ce2b443011e5a86125cd","first_computed_at":"2026-05-18T03:42:11.534004Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:42:11.534004Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"909D7LjNdwzzA8zFko9nyqQusoJl/FCuUiwVm58ncsPK3FxsGuOTGzksVco9a7GWmqDZDcouAQ7fVuS/BOu+Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:42:11.534787Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.2591","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a3727f752a2ec4957f694a7203ce5f345720d2f2b04a63116637d98b4a84b77b","sha256:be9aa7f5d941a18b8c4c509128ad01283c9524f1027efc3f2765b08530f368ed"],"state_sha256":"3adba01fbdcf631e4ae5d47f5427258d408055fd6fc76d1065b8b19722fcfc55"}