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Linear degenerations of flag varieties

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Abstract

Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A.

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Acknowledgements

X.F. is supported by the Alexander von Humboldt Foundation. X.F would like to thank G.C-I. for invitation to Sapienza-Università di Roma where part of this work is carried out. G.C-I., G.F. and M.R. were partially supported by the DFG priority program 1388 “Representation Theory”, in whose context this project has been initiated. EF was supported by the RSF-DFG grant 16-41-01013. G.C.I. was also supported by the italian FIRB program “Perspectives in Lie Theory” RBFR12RA9W.

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Authors and Affiliations

  1. Dipartimento di Matematica “G. Castelnuovo”, Sapienza Universitá di Roma, Piazzale Aldo Moro 5, 00185, Rome, Italy

    G. Cerulli Irelli

  2. Mathematical Institute, University of Cologne, Weyertal 86–90, 50931, Cologne, Germany

    X. Fang

  3. Department of Mathematics, National Research University Higher School of Economics, Vavilova Str. 7, Moscow, Russia, 117312

    E. Feigin

  4. Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Building 3, Moscow, 143026, Russia

    E. Feigin

  5. Institute for Algebra, Number Theory and Discrete Mathematics, Leibniz University Hannover, Welfengarten 1, 30167, Hannover, Germany

    G. Fourier

  6. Faculty of Mathematics, Ruhr-Universität Bochum, Universitätsstraße 150, 44780, Bochum, Germany

    M. Reineke

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  1. G. Cerulli Irelli
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  2. X. Fang
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  3. E. Feigin
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  4. G. Fourier
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  5. M. Reineke
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Correspondence to M. Reineke.

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Cerulli Irelli, G., Fang, X., Feigin, E. et al. Linear degenerations of flag varieties. Math. Z. 287, 615–654 (2017). https://doi.org/10.1007/s00209-016-1839-y

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  • DOI: https://doi.org/10.1007/s00209-016-1839-y

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