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Totally nonnegative Grassmannians, Grassmann necklaces, and quiver Grassmannians

Part of: Representation theory of rings and algebras

Published online by Cambridge University Press:  03 June 2022

Evgeny Feigin ,
Martina Lanini  and
Alexander Pütz
Evgeny Feigin
Affiliation:
Faculty of Mathematics, HSE University, Usacheva 6, Moscow 119048, Russia Center for Advanced Studies, Skolkovo Institute of Science and Technology, Bolshoy Boulevard 30, Building 1, Moscow 121205, Russia e-mail: evgfeig@gmail.com
Martina Lanini*
Affiliation:
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, Rome I-00133, Italy
Alexander Pütz
Affiliation:
Faculty of Mathematics, Ruhr-University Bochum, Universitätsstraße 150, Bochum 44780, Germany e-mail: alexander.puetz@ruhr-uni-bochum.de
*
e-mail: lanini@mat.uniroma2.it
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Abstract

Postnikov constructed a cellular decomposition of the totally nonnegative Grassmannians. The poset of cells can be described (in particular) via Grassmann necklaces. We study certain quiver Grassmannians for the cyclic quiver admitting a cellular decomposition, whose cells are naturally labeled by Grassmann necklaces. We show that the posets of cells coincide with the reversed cell posets of the cellular decomposition of the totally nonnegative Grassmannians. We investigate algebro-geometric and combinatorial properties of these quiver Grassmannians. In particular, we describe the irreducible components, study the action of the automorphism groups of the underlying representations, and describe the moment graphs. We also construct a resolution of singularities for each irreducible component; the resolutions are defined as quiver Grassmannians for an extended cyclic quiver.

Keywords

Quiver Grassmannianstotally nonnegative GrassmanniansGrassmann necklaces

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 4 , August 2023 , pp. 1076 - 1109
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

E.F. was partially supported by the Grant No. RSF 19-11-00056. The study has been partially funded within the framework of the HSE University Basic Research Program. M.L. acknowledges the PRIN2017 CUP E8419000480006, the Fondi di Ricerca Scientifica di Ateneo 2021 CUP E853C22001680005, and the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

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