[{"publication_status":"epub_ahead","publication_identifier":{"eissn":["1420-9020"],"issn":["1022-1824"]},"abstract":[{"text":"In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t2). Using these results together with results of Geiss, Leclerc and Schröer we give, when k is algebraically closed, a classification of pairs (Q, R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra Fq[t]/(tr). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.","lang":"eng"}],"intvolume":"        30","volume":30,"article_type":"original","date_created":"2024-02-04T23:00:53Z","author":[{"last_name":"Hausel","id":"4A0666D8-F248-11E8-B48F-1D18A9856A87","full_name":"Hausel, Tamás","first_name":"Tamás"},{"full_name":"Letellier, Emmanuel","last_name":"Letellier","first_name":"Emmanuel"},{"full_name":"Rodriguez-Villegas, Fernando","last_name":"Rodriguez-Villegas","first_name":"Fernando"}],"scopus_import":"1","day":"27","title":"Locally free representations of quivers over commutative Frobenius algebras","oa_version":"None","issue":"2","citation":{"ama":"Hausel T, Letellier E, Rodriguez-Villegas F. Locally free representations of quivers over commutative Frobenius algebras. <i>Selecta Mathematica</i>. 2024;30(2). doi:<a href=\"https://doi.org/10.1007/s00029-023-00914-2\">10.1007/s00029-023-00914-2</a>","ieee":"T. Hausel, E. Letellier, and F. Rodriguez-Villegas, “Locally free representations of quivers over commutative Frobenius algebras,” <i>Selecta Mathematica</i>, vol. 30, no. 2. Springer Nature, 2024.","short":"T. Hausel, E. Letellier, F. Rodriguez-Villegas, Selecta Mathematica 30 (2024).","ista":"Hausel T, Letellier E, Rodriguez-Villegas F. 2024. Locally free representations of quivers over commutative Frobenius algebras. Selecta Mathematica. 30(2), 20.","chicago":"Hausel, Tamás, Emmanuel Letellier, and Fernando Rodriguez-Villegas. “Locally Free Representations of Quivers over Commutative Frobenius Algebras.” <i>Selecta Mathematica</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00029-023-00914-2\">https://doi.org/10.1007/s00029-023-00914-2</a>.","apa":"Hausel, T., Letellier, E., &#38; Rodriguez-Villegas, F. (2024). Locally free representations of quivers over commutative Frobenius algebras. <i>Selecta Mathematica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00029-023-00914-2\">https://doi.org/10.1007/s00029-023-00914-2</a>","mla":"Hausel, Tamás, et al. “Locally Free Representations of Quivers over Commutative Frobenius Algebras.” <i>Selecta Mathematica</i>, vol. 30, no. 2, 20, Springer Nature, 2024, doi:<a href=\"https://doi.org/10.1007/s00029-023-00914-2\">10.1007/s00029-023-00914-2</a>."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"department":[{"_id":"TaHa"}],"article_number":"20","month":"01","quality_controlled":"1","date_updated":"2024-02-05T12:58:21Z","_id":"14930","type":"journal_article","doi":"10.1007/s00029-023-00914-2","article_processing_charge":"No","publisher":"Springer Nature","date_published":"2024-01-27T00:00:00Z","acknowledgement":"Special thanks go to Christof Geiss, Bernard Leclerc and Jan Schröer for explaining their work but also for sharing some unpublished results with us. We also thank the referee for many useful suggestions. We would like to thank Tommaso Scognamiglio for pointing out a mistake in the proof of Proposition 5.17 in an earlier version of the paper. We would like also to thank Alexander Beilinson, Bill Crawley-Boevey, Joel Kamnitzer, and Peng Shan for useful discussions.","status":"public","publication":"Selecta Mathematica","year":"2024"},{"project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"status":"public","publication":"Selecta Mathematica","acknowledgement":"First of all we would like to thank Andrei Okounkov for invaluable discussions, advises and sharing with us his fantastic viewpoint on modern quantum geometry. We are also grateful to D. Korb and Z. Zhou for their interest and comments. The work of A. Smirnov was supported in part by RFBR Grants under Numbers 15-02-04175 and 15-01-04217 and in part by NSF Grant DMS–2054527. The work of P. Koroteev, A.M. Zeitlin and A. Smirnov is supported in part by AMS Simons travel Grant. A. M. Zeitlin is partially supported by Simons Collaboration Grant, Award ID: 578501. Open access funding provided by Institute of Science and Technology (IST Austria).","date_published":"2021-08-30T00:00:00Z","external_id":{"isi":["000692795200001"]},"year":"2021","isi":1,"ddc":["530"],"quality_controlled":"1","publisher":"Springer Nature","doi":"10.1007/s00029-021-00698-3","article_processing_charge":"Yes (via OA deal)","type":"journal_article","date_updated":"2023-08-14T06:34:14Z","_id":"9998","language":[{"iso":"eng"}],"oa":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","issue":"5","citation":{"chicago":"Koroteev, Peter, Petr Pushkar, Andrey V. Smirnov, and Anton M. Zeitlin. “Quantum K-Theory of Quiver Varieties and Many-Body Systems.” <i>Selecta Mathematica</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00029-021-00698-3\">https://doi.org/10.1007/s00029-021-00698-3</a>.","ista":"Koroteev P, Pushkar P, Smirnov AV, Zeitlin AM. 2021. Quantum K-theory of quiver varieties and many-body systems. Selecta Mathematica. 27(5), 87.","apa":"Koroteev, P., Pushkar, P., Smirnov, A. V., &#38; Zeitlin, A. M. (2021). Quantum K-theory of quiver varieties and many-body systems. <i>Selecta Mathematica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00029-021-00698-3\">https://doi.org/10.1007/s00029-021-00698-3</a>","mla":"Koroteev, Peter, et al. “Quantum K-Theory of Quiver Varieties and Many-Body Systems.” <i>Selecta Mathematica</i>, vol. 27, no. 5, 87, Springer Nature, 2021, doi:<a href=\"https://doi.org/10.1007/s00029-021-00698-3\">10.1007/s00029-021-00698-3</a>.","ama":"Koroteev P, Pushkar P, Smirnov AV, Zeitlin AM. Quantum K-theory of quiver varieties and many-body systems. <i>Selecta Mathematica</i>. 2021;27(5). doi:<a href=\"https://doi.org/10.1007/s00029-021-00698-3\">10.1007/s00029-021-00698-3</a>","ieee":"P. Koroteev, P. Pushkar, A. V. Smirnov, and A. M. Zeitlin, “Quantum K-theory of quiver varieties and many-body systems,” <i>Selecta Mathematica</i>, vol. 27, no. 5. Springer Nature, 2021.","short":"P. Koroteev, P. Pushkar, A.V. Smirnov, A.M. Zeitlin, Selecta Mathematica 27 (2021)."},"month":"08","file":[{"checksum":"beadc5a722ffb48190e1e63ee2dbfee5","relation":"main_file","content_type":"application/pdf","access_level":"open_access","file_name":"2021_SelectaMath_Koroteev.pdf","success":1,"file_id":"10010","creator":"cchlebak","date_updated":"2021-09-13T11:31:34Z","date_created":"2021-09-13T11:31:34Z","file_size":584648}],"article_number":"87","department":[{"_id":"TaHa"}],"abstract":[{"text":"We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice.","lang":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"intvolume":"        27","has_accepted_license":"1","publication_status":"published","publication_identifier":{"issn":["1022-1824"],"eissn":["1420-9020"]},"file_date_updated":"2021-09-13T11:31:34Z","title":"Quantum K-theory of quiver varieties and many-body systems","oa_version":"Published Version","author":[{"full_name":"Koroteev, Peter","last_name":"Koroteev","first_name":"Peter"},{"first_name":"Petr","last_name":"Pushkar","id":"151DCEB6-9EC3-11E9-8480-ABECE5697425","full_name":"Pushkar, Petr"},{"first_name":"Andrey V.","last_name":"Smirnov","full_name":"Smirnov, Andrey V."},{"first_name":"Anton M.","full_name":"Zeitlin, Anton M.","last_name":"Zeitlin"}],"day":"30","scopus_import":"1","article_type":"original","date_created":"2021-09-12T22:01:22Z","volume":27}]