{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:LZ5JTBFDTAF7QDDH4CMIRGZD6V","short_pith_number":"pith:LZ5JTBFD","canonical_record":{"source":{"id":"2301.07221","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2023-01-17T23:10:52Z","cross_cats_sorted":[],"title_canon_sha256":"6e6aa83d0106d245e74048971af8f67dd75bfa81a14576298e30a6f17831c73b","abstract_canon_sha256":"7039074446f4e9604dd940d6396d65e6bcb27d807ebd5aa31b9361777e669f5b"},"schema_version":"1.0"},"canonical_sha256":"5e7a9984a3980bf80c67e098889b23f54bcfadc12271d5ea2f5ad90c1c06b810","source":{"kind":"arxiv","id":"2301.07221","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2301.07221","created_at":"2026-07-05T05:34:06Z"},{"alias_kind":"arxiv_version","alias_value":"2301.07221v1","created_at":"2026-07-05T05:34:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2301.07221","created_at":"2026-07-05T05:34:06Z"},{"alias_kind":"pith_short_12","alias_value":"LZ5JTBFDTAF7","created_at":"2026-07-05T05:34:06Z"},{"alias_kind":"pith_short_16","alias_value":"LZ5JTBFDTAF7QDDH","created_at":"2026-07-05T05:34:06Z"},{"alias_kind":"pith_short_8","alias_value":"LZ5JTBFD","created_at":"2026-07-05T05:34:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:LZ5JTBFDTAF7QDDH4CMIRGZD6V","target":"record","payload":{"canonical_record":{"source":{"id":"2301.07221","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2023-01-17T23:10:52Z","cross_cats_sorted":[],"title_canon_sha256":"6e6aa83d0106d245e74048971af8f67dd75bfa81a14576298e30a6f17831c73b","abstract_canon_sha256":"7039074446f4e9604dd940d6396d65e6bcb27d807ebd5aa31b9361777e669f5b"},"schema_version":"1.0"},"canonical_sha256":"5e7a9984a3980bf80c67e098889b23f54bcfadc12271d5ea2f5ad90c1c06b810","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:34:06.718330Z","signature_b64":"vHLSqbt54Y1U42hIvAvAo+z5WIAGjHbfs/SMVkXbFHkZIbWOPOkddozIMfbURkggm7o+HHZ/EqqJAmrqj/TmCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e7a9984a3980bf80c67e098889b23f54bcfadc12271d5ea2f5ad90c1c06b810","last_reissued_at":"2026-07-05T05:34:06.717875Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:34:06.717875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2301.07221","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":"2026-07-05T05:34:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jHvd1FmLKv9Ck2wfLaUiD7iFptiyuJDbqFXj9O7IHx5lAN3mizbluwo3bFIZX25/kTuU1o+60dyPZhoyqcocBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T11:20:17.823081Z"},"content_sha256":"fef7d47377cb0acd91ab59848e0f2427fc20e6754c1e955393ffb5eff1d9c2dc","schema_version":"1.0","event_id":"sha256:fef7d47377cb0acd91ab59848e0f2427fc20e6754c1e955393ffb5eff1d9c2dc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:LZ5JTBFDTAF7QDDH4CMIRGZD6V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Combinatorics of $\\mathbb{F}_1$-Representations of Pseudotree Quivers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Alex Sistko, Jaehoon Kim, Jaiung Jun","submitted_at":"2023-01-17T23:10:52Z","abstract_excerpt":"We investigate quiver representations over $\\mathbb{F}_1$. Coefficient quivers are combinatorial gadgets equivalent to $\\mathbb{F}_1$-representations of quivers. We focus on the case when the quiver $Q$ is a pseudotree. For such quivers, we first use the notion of coefficient quivers to provide a complete classification of asymptotic behaviors of indecomposable representations over $\\mathbb{F}_1$. Then, we prove some fundamental structural results about the Lie algebras associated to pseudotrees. Finally, we construct examples of $\\mathbb{F}_1$-representations $M$ of a quiver $Q$ by using cove"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.07221","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/2301.07221/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":"2026-07-05T05:34:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"njKOaJWoB690hcIy0VJ5JrIwO7NUWTIle1Q000bieBOw93ou6XEQHMmpAEnFY5AUow1QHUityBrHdtoEvpbeCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T11:20:17.823608Z"},"content_sha256":"8fad89bf9ff6efbd9a86478d93e8d4be09b720ce6302deb3403af5469e3f97aa","schema_version":"1.0","event_id":"sha256:8fad89bf9ff6efbd9a86478d93e8d4be09b720ce6302deb3403af5469e3f97aa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LZ5JTBFDTAF7QDDH4CMIRGZD6V/bundle.json","state_url":"https://pith.science/pith/LZ5JTBFDTAF7QDDH4CMIRGZD6V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LZ5JTBFDTAF7QDDH4CMIRGZD6V/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":"2026-08-10T11:20:17Z","links":{"resolver":"https://pith.science/pith/LZ5JTBFDTAF7QDDH4CMIRGZD6V","bundle":"https://pith.science/pith/LZ5JTBFDTAF7QDDH4CMIRGZD6V/bundle.json","state":"https://pith.science/pith/LZ5JTBFDTAF7QDDH4CMIRGZD6V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LZ5JTBFDTAF7QDDH4CMIRGZD6V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:LZ5JTBFDTAF7QDDH4CMIRGZD6V","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":"7039074446f4e9604dd940d6396d65e6bcb27d807ebd5aa31b9361777e669f5b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2023-01-17T23:10:52Z","title_canon_sha256":"6e6aa83d0106d245e74048971af8f67dd75bfa81a14576298e30a6f17831c73b"},"schema_version":"1.0","source":{"id":"2301.07221","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2301.07221","created_at":"2026-07-05T05:34:06Z"},{"alias_kind":"arxiv_version","alias_value":"2301.07221v1","created_at":"2026-07-05T05:34:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2301.07221","created_at":"2026-07-05T05:34:06Z"},{"alias_kind":"pith_short_12","alias_value":"LZ5JTBFDTAF7","created_at":"2026-07-05T05:34:06Z"},{"alias_kind":"pith_short_16","alias_value":"LZ5JTBFDTAF7QDDH","created_at":"2026-07-05T05:34:06Z"},{"alias_kind":"pith_short_8","alias_value":"LZ5JTBFD","created_at":"2026-07-05T05:34:06Z"}],"graph_snapshots":[{"event_id":"sha256:8fad89bf9ff6efbd9a86478d93e8d4be09b720ce6302deb3403af5469e3f97aa","target":"graph","created_at":"2026-07-05T05:34:06Z","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/2301.07221/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate quiver representations over $\\mathbb{F}_1$. Coefficient quivers are combinatorial gadgets equivalent to $\\mathbb{F}_1$-representations of quivers. We focus on the case when the quiver $Q$ is a pseudotree. For such quivers, we first use the notion of coefficient quivers to provide a complete classification of asymptotic behaviors of indecomposable representations over $\\mathbb{F}_1$. Then, we prove some fundamental structural results about the Lie algebras associated to pseudotrees. Finally, we construct examples of $\\mathbb{F}_1$-representations $M$ of a quiver $Q$ by using cove","authors_text":"Alex Sistko, Jaehoon Kim, Jaiung Jun","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2023-01-17T23:10:52Z","title":"On the Combinatorics of $\\mathbb{F}_1$-Representations of Pseudotree Quivers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.07221","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:fef7d47377cb0acd91ab59848e0f2427fc20e6754c1e955393ffb5eff1d9c2dc","target":"record","created_at":"2026-07-05T05:34:06Z","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":"7039074446f4e9604dd940d6396d65e6bcb27d807ebd5aa31b9361777e669f5b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2023-01-17T23:10:52Z","title_canon_sha256":"6e6aa83d0106d245e74048971af8f67dd75bfa81a14576298e30a6f17831c73b"},"schema_version":"1.0","source":{"id":"2301.07221","kind":"arxiv","version":1}},"canonical_sha256":"5e7a9984a3980bf80c67e098889b23f54bcfadc12271d5ea2f5ad90c1c06b810","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5e7a9984a3980bf80c67e098889b23f54bcfadc12271d5ea2f5ad90c1c06b810","first_computed_at":"2026-07-05T05:34:06.717875Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:34:06.717875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vHLSqbt54Y1U42hIvAvAo+z5WIAGjHbfs/SMVkXbFHkZIbWOPOkddozIMfbURkggm7o+HHZ/EqqJAmrqj/TmCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:34:06.718330Z","signed_message":"canonical_sha256_bytes"},"source_id":"2301.07221","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fef7d47377cb0acd91ab59848e0f2427fc20e6754c1e955393ffb5eff1d9c2dc","sha256:8fad89bf9ff6efbd9a86478d93e8d4be09b720ce6302deb3403af5469e3f97aa"],"state_sha256":"718d63accd3aaf4b72d4b76eafeef14c905b39a41a740f31356426336d53782b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Djkq00q0YwpWM+jfjt24nwJJTlq/Zkhclwi00MKHwW3Tta5TYXWkJHLxMJmMpeS0KEI/5RYwMhD1pR203IbYBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T11:20:17.830982Z","bundle_sha256":"727ca8e03f3b8a350660f634b47f20f981ddf0eec79ee1b68cf005c172ee48d3"}}