{"paper":{"title":"Quiver varieties and dual canonical bases","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"Dual canonical bases of quantum groups coincide with Berenstein-Greenstein double canonical bases.","cross_cats":["math.RT"],"primary_cat":"math.QA","authors_text":"Ming Lu, Xiaolong Pan","submitted_at":"2026-05-13T14:12:30Z","abstract_excerpt":"We survey some recent developments on the theory of dual canonical bases for quantum groups and $\\imath$quantum groups. The $\\imath$quiver algebras were introduced by Wang and the first author, which are used to give two realizations of quasi-split $\\imath$quantum groups of type ADE: one via the $\\imath$Hall algebras and the other via the quantum Grothendieck rings of Nakajima-Keller-Scherotzke quiver varieties. The geometric construction of the $\\imath$quantum groups produces their dual canonical bases with positivity, generalizing Qin's geometric realization of quantum groups of type ADE. Re"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we demonstrate that the dual canonical bases of quantum groups coincide with the double canonical bases defined by Berenstein and Greenstein, and resolve several conjectures therein.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the i-quiver algebras correctly provide the two realizations of quasi-split i-quantum groups (via i-Hall algebras and quantum Grothendieck rings of the specified quiver varieties) as stated in the prior introduction by Wang and the first author.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Dual canonical bases of quantum groups coincide with double canonical bases via i-quiver algebra and quiver variety constructions, with new proofs of positivity and braid group invariance.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Dual canonical bases of quantum groups coincide with Berenstein-Greenstein double canonical bases.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8bf828abbb6563c9c599877deb76fe9f93faa1e09921b061e8b085409faea87a"},"source":{"id":"2605.13578","kind":"arxiv","version":1},"verdict":{"id":"215d160e-ce97-4595-92fb-265d25905426","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:03:11.191038Z","strongest_claim":"we demonstrate that the dual canonical bases of quantum groups coincide with the double canonical bases defined by Berenstein and Greenstein, and resolve several conjectures therein.","one_line_summary":"Dual canonical bases of quantum groups coincide with double canonical bases via i-quiver algebra and quiver variety constructions, with new proofs of positivity and braid group invariance.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the i-quiver algebras correctly provide the two realizations of quasi-split i-quantum groups (via i-Hall algebras and quantum Grothendieck rings of the specified quiver varieties) as stated in the prior introduction by Wang and the first author.","pith_extraction_headline":"Dual canonical bases of quantum groups coincide with Berenstein-Greenstein double canonical bases."},"references":{"count":88,"sample":[{"doi":"","year":2021,"title":"Achar, Perverse sheaves and applications to representation theory, vol","work_id":"3b3e332c-0cd6-4228-b494-157c6eecf01d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"M. Auslander and R. Buchweitz,","work_id":"eff60a58-cac1-4bb5-80b1-85a0afed1237","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"H. Bao, J. Kujawa, Y. Li and W. Wang,","work_id":"95a03483-7d79-44d5-922e-a76152461b97","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"H. Bao, P. Shan, W. Wang and B. Webster,","work_id":"4b937d9c-09af-4ea4-a313-975e9828c5f6","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"H. Bao and W. Wang, A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs , Ast\\'erisque 402, 2018, vii+134pp","work_id":"b2366ee4-b25c-48f8-815c-5dffd9b4b256","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":88,"snapshot_sha256":"48c0742baf2156730a72ffdc4f84c342b0e42c794fc170e730bd3cb11ca4d0ad","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"}