Linear degenerations of flag varieties: partial flags, defining equations, and group actions
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Mathematische Zeitschrift
Linear degenerations of flag varieties: partial flags, defining equations, and group actions Giovanni Cerulli Irelli1 · Xin Fang2 · Evgeny Feigin3,4 · Ghislain Fourier5 · Markus Reineke6 Received: 19 February 2019 / Accepted: 21 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We continue, generalize and expand our study of linear degenerations of flag varieties from Cerulli Irelli et al. (Math Z 287(1–2):615–654, 2017). We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degenerate flag variety. We compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus. We also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus. Finally, we prove an analogue of the Borel–Weil theorem for the flat irreducible locus.
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Xin Fang [email protected] Giovanni Cerulli Irelli [email protected] Evgeny Feigin [email protected] Ghislain Fourier [email protected] Markus Reineke [email protected]
1
Dipartimento S.B.A.I., Sapienza Universitá di Roma, Via Scarpa 10, 00161 Rome, Italy
2
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
3
Department of Mathematics, National Research University Higher School of Economics, Usacheva Str. 6, Moscow 119048, Russia
4
Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Building 3, Moscow 143026, Russia
5
RWTH Aachen University, Pontdriesch 10-16, 52062 Aachen, Germany
6
Ruhr-Universität Bochum, Faculty of Mathematics, Universitätsstraße 150, 44780 Bochum, Germany
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G. Cerulli Irelli et al.
1 Introduction The theory of complex simple Lie groups and Lie algebras is known to be closely related to the representation theory of Dynkin quivers (see e.g. [1,11,14,18]). In this paper we use the following simple but powerful observation: any partial flag variety associated to the group SL N is isomorphic to a quiver Grassmannian for the equi-oriented type A quiver and suitably chosen representation and dimension vector. Varying the representation of the quiver and keeping the dimension vector fixed one gets degenerations of the flag varieties (see e.g. [12,13,15,16]). The goal of this paper is to study these degenerations, in particular, to describe the irreducible and flat irreducible loci. Let us formulate the setup and our results in more details. Let G = SL N (C) and let P be a parabolic subgroup of G with respect to the fixed Borel subgroup B. The quotient G/P is known to be isomorphic
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