{"title":"The monodromy of real Bethe vectors for the Gaudin model","authors":"Noah White","doi":"10.4171/JCA/2-3-3","DOIUrl":null,"url":null,"abstract":"The Bethe algebras for the Gaudin model act on the multiplicity space of tensor products of irreducible $ \\mathfrak{gl}_r $-modules and have simple spectrum over real points. This fact is proved by Mukhin, Tarasov and Varchenko who also develop a relationship to Schubert intersections over real points. We use an extension to $ \\overline{M}_{0,n+1}(\\mathbb{R}) $ of these Schubert intersections, constructed by Speyer, to calculate the monodromy of the spectrum of the Bethe algebras. We show this monodromy is described by the action of the cactus group $ J_n $ on tensor products of irreducible $ \\mathfrak{gl}_r $-crystals.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2015-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/2-3-3","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Algebra","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/JCA/2-3-3","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 8
Abstract
The Bethe algebras for the Gaudin model act on the multiplicity space of tensor products of irreducible $ \mathfrak{gl}_r $-modules and have simple spectrum over real points. This fact is proved by Mukhin, Tarasov and Varchenko who also develop a relationship to Schubert intersections over real points. We use an extension to $ \overline{M}_{0,n+1}(\mathbb{R}) $ of these Schubert intersections, constructed by Speyer, to calculate the monodromy of the spectrum of the Bethe algebras. We show this monodromy is described by the action of the cactus group $ J_n $ on tensor products of irreducible $ \mathfrak{gl}_r $-crystals.