Abstract
We study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and that their irreducible components are normal Cohen-Macaulay varieties with rational singularities.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Beauville, A., Laszlo, Y.: Conformal Blocks and Generalized Theta functions. Commun. Math. Phys. 164, 385–419 (1994)
Bongartz, K.: Grassmannians and varieties of modules. Unpublished manuscript (1997)
Carrell, J. B.: Torus Actions and Cohomology. In: Gamkrelidze, R. V. (ed.) Encyclopaedia of Mathematical Sciences (Invariant Theory and Algebraic Transformationgroups), pp 83–158. Springer, Berlin (2002)
Cerulli Irelli, G.: Quiver Grassmannians associated with string modules. J. Algebraic Comb. 33, 259–276 (2011)
Cerulli Irelli, G., Esposito, F., Franzen, H., Reineke, M: Cellular decomposition and algebraicity of cohomology for quiver Grassmannians. 1804.07736v3
Cerulli Irelli, G., Fang, X., Feigin, E., Fourier, G., Reineke, M.: Linear degenerations of flag varieties. Mathematische Zeitschrift 287, 615–654 (2017)
Cerulli Irelli, G., Feigin, E., Reineke, M.: Quiver Grassmannians and degenerate flag varieties. Algebra Number Theory 6, 165–194 (2012)
Crawley-Boevey, W.: Maps between representations of zero-relation algebras. J. Algebra 126, 259–263 (1989)
De Concini, C., Lusztig, G., Procesi, C.: Homology of the zero-set of nilpotent vector field on a flag manifold. J. Amer. Math. Soc. 1, 15–34 (1988)
Fang, X., Fourier, G.: Torus fixed points in Schubert varieties and normalized median Genocchi numbers. Séminaire Lotharingien de Combinatoire f75, Article B75f. (2016)
Fang, X., Fourier, G., Littelmann, P.: On toric degenerations of flag varieties. Representation Theory - Current Trends and Perspectives, EMS Series of Congress Reports, pp. 187–232 (2016)
Fedotov, S.: Framed Moduli and Grassmannians of submodules. Trans. Amer. Math. Soc. 365, 4153–4179 (2013)
Feigin, E.: Degenerate flag varieties and the median Genocchi numbers. Math. Re. Lett. 18, 1163–1178 (2011)
Feigin, E., Finkelberg, M., Reineke, M.: Degenerate affine Grassmannians and loop quivers. Kyoto J. Math. 57, 445–474 (2017)
Gabriel, P.: Unzerlegbare Darstellungen I. Manuscripta Math. 6, 71–103 (1972)
Görtz, U.: On the flatness of models of certain Shimura varieties of PEL-type. Math. Ann. 321, 689–727 (2001)
Görtz, U.: Affine Springer Fibres and Affine Deligne-Lusztig Varieties. In: Schmitt, A. (ed.) Affine Flag Manifolds and Principal Bundles, pp 1–50. Trends in Mathematics, Springer (2010)
Haupt, N.: Euler characteristics and geometric properties of quiver Grassmannians. Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn (2011). http://hss.ulb.uni-bonn.de/2011/2673/2673.pdf
Haupt, N.: Euler Characteristics of Quiver Grassmannians and Ringel-Hall Algebras of String Algebras. Algebras Represent. Theory 15, 755–793 (2012)
Hubery, A.: Ringel-Hall Algebras of Cyclic Quivers. São Paulo J. Math. Sci. 4, 351–398 (2010)
Kac, V., Peterson, D.: Lectures on the infinite wedge representation and the MKP hierarchy. Seminaire de Math. Superieures, Les Presses de L’Universite de Montreal f102, 141–186 (1986)
Kempf, G.: Toroidal Embeddings I. Springer Lecture Notes 339, Springer, Berlin (1973)
Kempken, G.: Eine Darstellung des Köchers \(\widetilde {A}_{k}\). Bonner Mathematische Schriften, Nr. 137, Bonn (1982)
Kirillov, A. Jr: Quiver Representations and Quiver Varieties. Graduate studies in mathematics, vol. 174, American Mathematical Society, Providence (2016)
Kumar, S.: Kac-Moody Groups, their Flag Varieties and Representation Theory. Progress in Mathematics, vol. 204, Birkhäuser, Boston (2002)
Pütz, A.: Degenerate Affine Flag Varieties and Quiver Grassmannians. Dissertation, Ruhr-Universität Bochum (2019). https://hss-opus.ub.rub.de/opus4/frontdoor/deliver/index/docId/6576/file/diss.pdf
Reineke, M.: The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli. Inventiones Math. 152(2), 349–368 (2003)
Reineke, M.: Framed quiver moduli, cohomology, and quantum groups. J. Algebra 320, 94–115 (2008)
Sauter, J.: Cell Decompositions of Quiver Flag Varieties for Nilpotent Representations of the Cyclic Quiver. Algebras Represent. Theory 20, 1323–1340 (2017)
Schiffler, R.: Quiver Representations. CMS Books in Mathematics. Springer, Cham (2014)
Schofield, A.: General representations of quivers. Proc. Lond. Math. Soc. 65, 46–64 (1992)
Stein, W.A., et al.: Sage Mathematics Software (Version 8.1). The Sage Development Team (2017). http://www.sagemath.org
Acknowledgements
This research was funded by the DFG/RSF project ”Geometry and representation theory at the interface of Lie algebras and quivers”. Furthermore, I acknowledge the PRIN2017 CUP E8419000480006, and the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. I want to thank M. Reineke for many very inspiring discussions during the preparation of this work and M. Lanini for very helpful suggestions to correct a mistake in a previous version. Moreover I am grateful to an anonymous referee for very useful suggestions to optimise the structure of this article.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interests
The author declares that he has no conflict of interest.
Additional information
Presented by: Peter Littelmann
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Availability of data and material
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Code availability
Not applicable.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Pütz, A. Degenerate Affine Flag Varieties and Quiver Grassmannians. Algebr Represent Theor 25, 91–119 (2022). https://doi.org/10.1007/s10468-020-10012-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-020-10012-y
Keywords
- Linear degenerations
- Finite approximations
- Equioriented cycle
- Rational singularities
- Grand Motzkin paths
- Affine Dellac configurations