J. Kamnitzer, P. Tingley, Ben Webster, Alex Weekes, Oded Yacobi
Truncated shifted Yangians are a family of algebras which are natural quantizations of slices in the affine Grassmannian. We study the highest weight representations of these algebras. In particular, we conjecture that the possible highest weights for these algebras are described by product monomial crystals, certain natural subcrystals of Nakajima's monomials. We prove this conjecture in type A. We also place our results in the context of symplectic duality and prove a conjecture of Hikita in this situation.
{"title":"Highest weights for truncated shifted Yangians and product monomial crystals","authors":"J. Kamnitzer, P. Tingley, Ben Webster, Alex Weekes, Oded Yacobi","doi":"10.4171/JCA/32","DOIUrl":"https://doi.org/10.4171/JCA/32","url":null,"abstract":"Truncated shifted Yangians are a family of algebras which are natural quantizations of slices in the affine Grassmannian. We study the highest weight representations of these algebras. In particular, we conjecture that the possible highest weights for these algebras are described by product monomial crystals, certain natural subcrystals of Nakajima's monomials. We prove this conjecture in type A. We also place our results in the context of symplectic duality and prove a conjecture of Hikita in this situation.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2015-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/32","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70870606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"The monodromy of real Bethe vectors for the Gaudin model","authors":"Noah White","doi":"10.4171/JCA/2-3-3","DOIUrl":"https://doi.org/10.4171/JCA/2-3-3","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":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2015-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/2-3-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70870960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the Gaussent--Littelmann root operators on galleries can be expressed purely in terms of retractions of a (Bruhat-Tits) building. In addition we establish a connection to the root datum at infinity.
{"title":"Root operators, root groups and retractions","authors":"Petra Schwer","doi":"10.4171/JCA/2-3-1","DOIUrl":"https://doi.org/10.4171/JCA/2-3-1","url":null,"abstract":"We prove that the Gaussent--Littelmann root operators on galleries can be expressed purely in terms of retractions of a (Bruhat-Tits) building. In addition we establish a connection to the root datum at infinity.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2015-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/2-3-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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