{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:QSMLZQ6RX522REK4VQRZM7W225","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":"cdc552117f7de080bef663f9e6af6b9fee82e4b695636c456e781ff893fef14e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T12:26:20Z","title_canon_sha256":"ea71bb12afc94ef4e8a4a9dcf634eaa3f11908cc68dcf86c9c04453230a6ef35"},"schema_version":"1.0","source":{"id":"2605.13432","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13432","created_at":"2026-05-18T02:44:47Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13432v1","created_at":"2026-05-18T02:44:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13432","created_at":"2026-05-18T02:44:47Z"},{"alias_kind":"pith_short_12","alias_value":"QSMLZQ6RX522","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"QSMLZQ6RX522REK4","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"QSMLZQ6R","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:aaf668220496b361497d2b49456d713a910ecdf524a3eb01b8aa96031b5714bc","target":"graph","created_at":"2026-05-18T02:44:47Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"inhomogeneous q-Whittaker polynomials (in countably many variables) form a basis of certain commutative ring extending the ring of symmetric functions to a subring of its completion"},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The inhomogeneous q-Whittaker polynomials are defined such that they simultaneously extend q-Whittaker and stable Grothendieck polynomials while satisfying the algebraic relations needed for the ring to be commutative and for the basis property to hold in the completion."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Inhomogeneous q-Whittaker polynomials form a basis for an extended commutative ring of symmetric functions and admit positive specializations related to a subset of Macdonald-positive ones, yielding associated probability distributions."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Inhomogeneous q-Whittaker polynomials form a basis for a commutative ring extending the symmetric functions to a subring of its completion."}],"snapshot_sha256":"4c3cd39b3220364c8e43b72741c0d0c420dd9df79ed0b74b5022e5a856925882"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study inhomogeneous $q$-Whittaker polynomials which extend both $q$-Whittaker and stable Grothendieck polynomials. We prove that inhomogeneous $q$-Whittaker polynomials (in countably many variables) form a basis of certain commutative ring extending the ring of symmetric functions to a subring of its completion. We then describe positive specializations of that ring and relate them with a subset of Macdonald-positive specializations of the ring of symmetric functions. We also show some related probability distributions obtained from positive specializations of inhomogeneous $q$-Whittaker po","authors_text":"Ajeeth Gunna, Damir Yeliussizov","cross_cats":[],"headline":"Inhomogeneous q-Whittaker polynomials form a basis for a commutative ring extending the symmetric functions to a subring of its completion.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T12:26:20Z","title":"Inhomogeneous $q$-Whittaker polynomials II: ring theorem and positive specializations"},"references":{"count":22,"internal_anchors":9,"resolved_work":22,"sample":[{"cited_arxiv_id":"1410.0976","doi":"","is_internal_anchor":true,"ref_index":1,"title":"On a family of symmetric rational functions","work_id":"9054ae2c-46be-45ea-a14c-4ed206915d4f","year":2017},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Inhomogeneous spinq-Whittaker polynomials.Annales de la Faculté des sciences de Toulouse : Mathématiques, 33(1):1–68, 2024.arXiv:2104.01415","work_id":"87152918-a7b2-46b8-9f2a-7801c7843a53","year":2024},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Cambridge Studies in Advanced Mathematics","work_id":"66c9363f-2f3a-4a44-97b0-4ff89c49a197","year":2016},{"cited_arxiv_id":"1701.06292","doi":"","is_internal_anchor":true,"ref_index":4,"title":"Spin $q$-Whittaker polynomials","work_id":"240b60ca-71f9-4765-9d9c-5e84e2a0beec","year":2021},{"cited_arxiv_id":"math/0004137","doi":"","is_internal_anchor":true,"ref_index":5,"title":"A Littlewood-Richardson rule for the K-theory of Grassmannians","work_id":"73d1279d-66b7-4eb3-83e8-0351914d24c9","year":2002}],"snapshot_sha256":"6a417ed2dc9f8c7f54600dc237ea2136f7edc48562c1d5f788983927d8319cf6"},"source":{"id":"2605.13432","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T18:06:37.865082Z","id":"86928ff0-3437-4e41-ba52-d8791f2e51bf","model_set":{"reader":"grok-4.3"},"one_line_summary":"Inhomogeneous q-Whittaker polynomials form a basis for an extended commutative ring of symmetric functions and admit positive specializations related to a subset of Macdonald-positive ones, yielding associated probability distributions.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Inhomogeneous q-Whittaker polynomials form a basis for a commutative ring extending the symmetric functions to a subring of its completion.","strongest_claim":"inhomogeneous q-Whittaker polynomials (in countably many variables) form a basis of certain commutative ring extending the ring of symmetric functions to a subring of its completion","weakest_assumption":"The inhomogeneous q-Whittaker polynomials are defined such that they simultaneously extend q-Whittaker and stable Grothendieck polynomials while satisfying the algebraic relations needed for the ring to be commutative and for the basis property to hold in the completion."}},"verdict_id":"86928ff0-3437-4e41-ba52-d8791f2e51bf"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2d7348dac1b560bbacb0f0ec6f24b2f8b6c397e0f85e418b06c95a1f62989151","target":"record","created_at":"2026-05-18T02:44:47Z","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":"cdc552117f7de080bef663f9e6af6b9fee82e4b695636c456e781ff893fef14e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T12:26:20Z","title_canon_sha256":"ea71bb12afc94ef4e8a4a9dcf634eaa3f11908cc68dcf86c9c04453230a6ef35"},"schema_version":"1.0","source":{"id":"2605.13432","kind":"arxiv","version":1}},"canonical_sha256":"8498bcc3d1bf75a8915cac23967edad76f6d656a9ff8f9adcf15b702f36bfd0c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8498bcc3d1bf75a8915cac23967edad76f6d656a9ff8f9adcf15b702f36bfd0c","first_computed_at":"2026-05-18T02:44:47.153483Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:47.153483Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sx/6fQf0C5X0nFWJ10uZJPXgViAjA3/P6+ct4sLyBlyAOUie6dNys1WdgD7YU091v4ogjiuKJI5Sb/VU4vL5BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:47.153896Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.13432","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d7348dac1b560bbacb0f0ec6f24b2f8b6c397e0f85e418b06c95a1f62989151","sha256:aaf668220496b361497d2b49456d713a910ecdf524a3eb01b8aa96031b5714bc"],"state_sha256":"3573b4c9c30fa301e816cb2cc53af45f14ca02b9155a053f0e14ee786abeec0d"}