{"title":"The Berenstein–Kirillov group and cactus groups","authors":"Michael Chmutov, Max Glick, P. Pylyavskyy","doi":"10.4171/jca/36","DOIUrl":"https://doi.org/10.4171/jca/36","url":null,"abstract":"Berenstein and Kirillov have studied the action of Bender-Knuth moves on semistandard tableaux. Losev has studied a cactus group action in Kazhdan-Lusztig theory; in type $A$ this action can also be identified in the work of Henriques and Kamnitzer. We establish the relationship between the two actions. We show that the Berenstein-Kirillov group is a quotient of the cactus group. We use this to derive previously unknown relations in the Berenstein-Kirillov group. We also determine precise implications between subsets of relations in the two groups, which yields a presentation for cactus groups in terms of Bender-Knuth generators.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2016-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jca/36","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871072","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}
J. Kamnitzer, P. Tingley, Ben Webster, Alex Weekes, Oded Yacobi
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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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