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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Combinatorial characterization of the weight monoids of smooth affine spherical varieties


Authors: Guido Pezzini and Bart Van Steirteghem
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 14M27, 20G05
DOI: https://doi.org/10.1090/tran/7785
Published electronically: February 25, 2019
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Abstract: Let $ G$ be a connected reductive group, and let $ X$ be a smooth affine spherical $ G$-variety, both defined over the complex numbers. A well-known theorem of I. Losev's says that $ X$ is uniquely determined by its weight monoid, which is the set of irreducible representations of $ G$ that occur in the coordinate ring of $ X$. In this paper, we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties.


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Additional Information

Guido Pezzini
Affiliation: Dipartimento di Matematica “Guido Castelnuovo”, “Sapienza” Università di Roma, Rome, Italy
Email: pezzini@mat.uniroma1.it

Bart Van Steirteghem
Affiliation: Department Mathematik, Emmy–Noether–Zentrum, FAU Erlangen-Nürnberg, Erlangen, Germany; and Department of Mathematics, Medgar Evers College—City University of New York, Brooklyn, New York
Email: bartvs@mec.cuny.edu

DOI: https://doi.org/10.1090/tran/7785
Received by editor(s): April 4, 2017
Received by editor(s) in revised form: August 7, 2018, and October 28, 2018
Published electronically: February 25, 2019
Additional Notes: The first author was partially supported by the DFG Schwerpunktprogramm 1388—Darstellungstheorie.
The second author received support from the City University of New York PSC-CUNY Research Award Program, and from the National Science Foundation through grant DMS 1407394. He also thanks Medgar Evers College for his 2016-17 Fellowship Award.
Article copyright: © Copyright 2019 American Mathematical Society