Selecta Mathematica

, Volume 24, Issue 5, pp 4781–4810 | Cite as

Affine PBW bases and affine MV polytopes

  • Dinakar MuthiahEmail author
  • Peter Tingley


We show how affine PBW bases can be used to construct affine MV polytopes, and that the resulting objects agree with the affine MV polytopes recently constructed using either preprojective algebras or KLR algebras. To do this we first generalize work of Beck–Chari–Pressley and Beck–Nakajima to define affine PBW bases for arbitrary convex orders on positive roots. Our results describe how affine PBW bases for different convex orders are related, answering a question posed by Beck and Nakajima.


Affine MV polytope Quantum affine algebra PBW basis Crystal 

Mathematics Subject Classification

Primary 17B37 Secondary 17B67 81R10 


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We thank Joel Kamnitzer and Peter McNamara for many useful discussions. D.M. was supported by a PIMS Postdoctoral Fellowship. P.T. was partially supported by NSF Grant DMS-1265555.


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Massachusetts AmherstAmherstUSA
  2. 2.Department of Mathematics and StatisticsLoyola University ChicagoChicagoUSA

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