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Firstly, motivated by known formulae for values of Deligne-Lusztig characters of finite reductive groups, we propose a formula for the values of the unipotent characters of ${\\mathbb{G}}(q)$ on the elements of $S$. Using this, we explicitly list the unipotent character values of the ${\\mathbb{Z}}_2$-spets $G_{24}(q)$ related to the Benson-Solomon fusion system Sol$(q)$.\n  Secondly, when $\\ell > 2$ is a very good prime for ${\\mathbb{G"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2507.08502","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.RT","submitted_at":"2025-07-11T11:24:38Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"e4ec6a0d2b1f551d836bf127d9535fec885ce317f5f7bff5bcf608d508fb956b","abstract_canon_sha256":"13744e86d186876961d95716451c37030c389b145e5879f45f56773cf136bba5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:35:41.214227Z","signature_b64":"gOQpCF/FslVR4YyWJd7Mq81lJnQXyqSWHvR1qpBKrp/B1y8Q2DSXTX5gE/xwTZn93YHaV4QJQQ7jg9ZurNfuBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14652a42a78f1e9e52cbd05d09f6d7e69c7eeeb74a2b00a5d18cc724a6069c1c","last_reissued_at":"2026-07-05T11:35:41.213789Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:35:41.213789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Partial character tables for $\\mathbb{Z}_\\ell$-spetses","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RT","authors_text":"Gunter Malle, Jason Semeraro, Radha Kessar","submitted_at":"2025-07-11T11:24:38Z","abstract_excerpt":"Let ${\\mathbb{G}}$ be a simply connected ${\\mathbb{Z}}_\\ell$-spets, let $q$ be a prime power, prime to $\\ell$ and let $S$ be the underlying Sylow $\\ell$-subgroup. Firstly, motivated by known formulae for values of Deligne-Lusztig characters of finite reductive groups, we propose a formula for the values of the unipotent characters of ${\\mathbb{G}}(q)$ on the elements of $S$. Using this, we explicitly list the unipotent character values of the ${\\mathbb{Z}}_2$-spets $G_{24}(q)$ related to the Benson-Solomon fusion system Sol$(q)$.\n  Secondly, when $\\ell > 2$ is a very good prime for ${\\mathbb{G"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.08502","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/2507.08502/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2507.08502","created_at":"2026-07-05T11:35:41.213844+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.08502v1","created_at":"2026-07-05T11:35:41.213844+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.08502","created_at":"2026-07-05T11:35:41.213844+00:00"},{"alias_kind":"pith_short_12","alias_value":"CRSSUQVHR4PJ","created_at":"2026-07-05T11:35:41.213844+00:00"},{"alias_kind":"pith_short_16","alias_value":"CRSSUQVHR4PJ4UWL","created_at":"2026-07-05T11:35:41.213844+00:00"},{"alias_kind":"pith_short_8","alias_value":"CRSSUQVH","created_at":"2026-07-05T11:35:41.213844+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2607.00515","citing_title":"Harish-Chandra theories, Ennola $d$-ality and Rouquier blocks for spetses","ref_index":13,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CRSSUQVHR4PJ4UWL2BOQT5WX42","json":"https://pith.science/pith/CRSSUQVHR4PJ4UWL2BOQT5WX42.json","graph_json":"https://pith.science/api/pith-number/CRSSUQVHR4PJ4UWL2BOQT5WX42/graph.json","events_json":"https://pith.science/api/pith-number/CRSSUQVHR4PJ4UWL2BOQT5WX42/events.json","paper":"https://pith.science/paper/CRSSUQVH"},"agent_actions":{"view_html":"https://pith.science/pith/CRSSUQVHR4PJ4UWL2BOQT5WX42","download_json":"https://pith.science/pith/CRSSUQVHR4PJ4UWL2BOQT5WX42.json","view_paper":"https://pith.science/paper/CRSSUQVH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.08502&json=true","fetch_graph":"https://pith.science/api/pith-number/CRSSUQVHR4PJ4UWL2BOQT5WX42/graph.json","fetch_events":"https://pith.science/api/pith-number/CRSSUQVHR4PJ4UWL2BOQT5WX42/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CRSSUQVHR4PJ4UWL2BOQT5WX42/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CRSSUQVHR4PJ4UWL2BOQT5WX42/action/storage_attestation","attest_author":"https://pith.science/pith/CRSSUQVHR4PJ4UWL2BOQT5WX42/action/author_attestation","sign_citation":"https://pith.science/pith/CRSSUQVHR4PJ4UWL2BOQT5WX42/action/citation_signature","submit_replication":"https://pith.science/pith/CRSSUQVHR4PJ4UWL2BOQT5WX42/action/replication_record"}},"created_at":"2026-07-05T11:35:41.213844+00:00","updated_at":"2026-07-05T11:35:41.213844+00:00"}