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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.

References

Carrell, J. B., Torus actions and cohomology . In: Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action, Encyclopaedia of Mathematical Sciences, 131, Springer, Berlin–Heidelberg, 2002.Google Scholar
Cerulli Irelli, G., Quiver Grassmannians associated with string modules. J. Algebraic Comb. 33(2011), 259–276.CrossRefGoogle Scholar
Cerulli Irelli, G., Three lectures on quiver Grassmannians . In: Št’ovíček, J. and Trlifaj, J. (eds.), Representation theory and beyond, Contemporary Mathematics, 758, American Mathematical Society, Providence, RI, 2020, pp. 57–88.CrossRefGoogle Scholar
Cerulli Irelli, G., Fang, X., Feigin, E., Fourier, G., and Reineke, M., Linear degenerations of flag varieties . Math. Z. 287(2017), no. 1, 615–654.CrossRefGoogle Scholar
Cerulli Irelli, G., Feigin, E., and Reineke, M., Quiver Grassmannians and degenerate flag varieties . Algebra Number Theory 6(2012), no. 1, 165–194.CrossRefGoogle Scholar
Cerulli Irelli, G., Feigin, E., and Reineke, M., Desingularization of quiver Grassmannians for Dynkin quivers . Adv. Math. 245(2013), 182–207.CrossRefGoogle Scholar
Cerulli Irelli, G., Feigin, E., and Reineke, M., Homological approach to the Hernandez–Leclerc construction and quiver varieties . Represent. Theory 18(2014), 1–14.CrossRefGoogle Scholar
Cerulli Irelli, G. and Lanini, M., Degenerate flag varieties of type A and C are Schubert varieties . Int. Math. Res. Not. IMRN 15(2015), 6353–6374.CrossRefGoogle Scholar
Feigin, E., Degenerate flag varieties and the median Genocchi numbers . Math. Res. Lett. 18(2011), no. 6, 1–16.CrossRefGoogle Scholar
Feigin, E., ${G}_a^M$ degeneration of flag varieties. Sel. Math. New Ser. 18(2012), no. 3, 513–537.CrossRefGoogle Scholar
Feigin, E. and Finkelberg, M., Degenerate flag varieties of type A: Frobenius splitting and BW theorem . Math. Z. 275(2013), nos. 1–2, 55–77.CrossRefGoogle Scholar
Feigin, E., Finkelberg, M., and Littelmann, P., Symplectic degenerate flag varieties. Can. J. Math. 66(2014), 1250–1286.CrossRefGoogle Scholar
Gantmacher, F. R. and Krein, M. G., Sur les matrices completement nonnegatives et oscillatoires. Compos. Math. 4(1937), 445–4276.Google Scholar
Gel’fand, I., Goresky, M., MacPherson, R., and Serganova, V., Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. Math. 63(1987), 301–316.CrossRefGoogle Scholar
Keller, B. and Scherotzke, S., Desingularizations of quiver Grassmannians via graded quiver varieties . Adv. Math. 256(2014), 318–347.CrossRefGoogle Scholar
Kempken, G., Eine Darstellung des Köchers ${\widetilde{A}}_k$ . Bonner Mathematische Schriften, Nr. 137, Bonn, 1982.Google Scholar
Knutson, A., The cyclic Bruhat decomposition of ${\mathrm{Gr}}_k\left({\mathbb{C}}^n\right)$ from the affine Bruhat decomposition of $\mathrm{AFla}{g}_k^{\circ }$ . Talk at Bert Kostant’s 80th birthday conference, 2008, http://pi.math.cornell.edu/allenk/positroid.pdf Google Scholar
Knutson, A., Lam, T., and Speyer, D. E., Positroid varieties: juggling and geometry . Compos. Math. 149(2013), no. 10, 1710–1752.CrossRefGoogle Scholar
Knutson, A., Lam, T., and Speyer, D. E., Projections of Richardson varieties . J. Reine Angew. Math. 687(2014), 133–157.CrossRefGoogle Scholar
Lam, T., Totally nonnegative Grassmannian and Grassmann polytopes . In: Current developments in mathematics 2014, International Press, Somerville, MA, 2016, pp. 51–152.Google Scholar
Lanini, M. and Pütz, A., GKM-theory for Torus actions on cyclic quiver Grassmannians. Preprint, 2020. arXiv:2008.13138 Google Scholar
Lanini, M. and Pütz, A., Permutation actions on quiver Grassmannians for the equioriented cycle via GKM-Theory. Preprint, 2021. arXiv:2105.10122 Google Scholar
Lusztig, G., Total positivity in reductive groups . In: Lie theory and geometry, Progress in Mathematics, 123, Birkhäuser Boston, Boston, MA, 1994, pp. 531–568.CrossRefGoogle Scholar
Lusztig, G., Total positivity in partial flag manifolds . Represent. Theory 2(1998), 70–78.CrossRefGoogle Scholar
Lusztig, G., Introduction to total positivity . In: Hilgert, J., Lawson, J. D., Neeb, K.-H., and Vinberg, E. B. (eds.), Positivity in Lie theory: open problems, De Gruyter Exp. Math., 26, de Gruyter, Berlin, 1998, pp. 133–145.Google Scholar
Postnikov, A., Total positivity, Grassmannians, and networks. Preprint, 2006. arXiv:math/0609764. http://math.mit.edu/apost/papers/tpgrass.pdf Google Scholar
Pütz, A., Degenerate affine flag varieties and quiver Grassmannians . Algebr. Represent. Theory 25(2020), 91–119. https://doi.org/10.1007/s10468-020-10012-y CrossRefGoogle Scholar
Pütz, A., Degenerate affine flag varieties and quiver Grassmannians, Ph.D. thesis, Ruhr-Universität Bochum, 2019. https://doi.org/10.13154/294-6576 CrossRefGoogle Scholar
Reineke, M., Framed quiver moduli, cohomology, and quantum groups . J. Algebra 320(2008), no. 1, 94–115.CrossRefGoogle Scholar
Reineke, M., Every projective variety is a quiver Grassmannian . Algebr. Represent. Theory 16(2013), 1313–1314.CrossRefGoogle Scholar
Rietsch, K., An algebraic cell decomposition of the nonnegative part of a flag variety . J. Algebra 213(1999), no. 1, 144–154.CrossRefGoogle Scholar
Rietsch, K., Closure relations for totally nonnegative cells in $G/ P$ . Math. Res. Lett. 13(2006), 775–786.CrossRefGoogle Scholar
Scherotzke, S., Desingularisation of quiver Grassmannians via Nakajima categories . Algebr. Represent. Theory 20(2017), 231–243.CrossRefGoogle Scholar
Schiffler, R., Quiver representations, CMS Books in Mathematics, Springer, Cham, 2014.CrossRefGoogle Scholar
Schoenberg, I. J., On totally positive functions, Laplace integrals and entire functions of the Laguerre–Polya–Schur type . Proc. Natl. Acad. Sci. USA 33(1947), 11–17.CrossRefGoogle ScholarPubMed
Schofield, A., General representations of quivers. Proc. Lond. Math. Soc. (3) 65(1992), no. 1, 46–64.CrossRefGoogle Scholar
Williams, L., Enumeration of totally positive Grassmann cells . Adv. Math. 190(2005), no. 2, 319–342.CrossRefGoogle Scholar