{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T23:45:58Z","timestamp":1771544758574,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We study the cluster variables and \"imaginary\" elements of the semicanonical basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$.  A closed formula for the Laurent expansions of these elements was given by P.Caldero and the author.  As a by-product, there was given a combinatorial interpretation of the Laurent polynomials in question, equivalent to the one obtained by G.Musiker and J.Propp.  The original argument by P.Caldero and the author used a geometric interpretation of the Laurent polynomials due to P.Caldero and F.Chapoton.  This note provides a quick, self-contained and completely elementary alternative proof of the same results.<\/jats:p>","DOI":"10.37236\/1005","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:52:03Z","timestamp":1578718323000},"source":"Crossref","is-referenced-by-count":9,"title":["Semicanonical Basis Generators of the Cluster Algebra of Type $A_1^{(1)}$"],"prefix":"10.37236","volume":"14","author":[{"given":"Andrei","family":"Zelevinsky","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2007,1,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1n4\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1n4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:04:43Z","timestamp":1579320283000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v14i1n4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1,19]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2007,1,3]]}},"URL":"https:\/\/doi.org\/10.37236\/1005","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1,19]]},"article-number":"N4"}}