Totally nonnegative Grassmannians, Grassmann necklaces and quiver Grassmannians

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.

[1]  A. Schofield.  General Representations of Quivers , 1992 .

[2]  J. Hilgert,et al.  Introduction to total positivity , 1998 .

[3]  Framed quiver moduli, cohomology, and quantum groups , 2004, math/0411101.

[4]  E. Feigin,et al.  Quiver Grassmannians and degenerate flag varieties , 2011, 1106.2399.

[5]  Y. Kodama,et al.  KP solitons and total positivity for the Grassmannian , 2011, 1106.0023.

[6]  K. Rietsch.  An Algebraic Cell Decomposition of the Nonnegative Part of a Flag Variety , 1997, alg-geom/9709035.

[7]  Lauren K. Williams,et al.  Enumeration of totally positive Grassmann cells , 2003, math/0307271.

[8]  L. Williams,et al.  Discrete Morse theory for totally non-negative flag varieties☆ , 2008, 0810.4314.

[9]  E. Feigin,et al.  Desingularization of quiver Grassmannians for Dynkin quivers , 2012, 1209.3960.

[10]  M. Finkelberg,et al.  Degenerate flag varieties of type A: Frobenius splitting and BW theorem , 2011, 1103.1491.

[11]  Sarah Scherotzke.  Desingularization of Quiver Grassmannians via Nakajima Categories , 2017 .

[12]  David E Speyer,et al.  Matching polytopes, toric geometry, and the totally non-negative Grassmannian , 2009 .

[13]  Giovanni Cerulli Irelli,et al.  Quiver Grassmannians associated with string modules , 2009, 0910.2592.

[14]  I. Gelfand,et al.  Combinatorial Geometries, Convex Polyhedra, and Schubert Cells , 1987 .

[15]  M. Reineke,et al.  Linear degenerations of flag varieties , 2016, Mathematische Zeitschrift.

[16]  Closure relations for totally nonnegative cells in G/P , 2005, math/0509137.

[17]  F. Gantmakher,et al.  Sur les matrices complètement non négatives et oscillatoires , 1937 .

[18]  Martina Lanini,et al.  Permutation actions on Quiver Grassmannians for the equioriented cycle via GKM-theory , 2021, Journal of Algebraic Combinatorics.

[19]  Suho Oh,et al.  Positroids and Schubert matroids , 2008, J. Comb. Theory, Ser. A.

[20]  On the quiver Grassmannian in the acyclic case , 2006, math/0611074.

[21]  J. Hilgert.  Positivity in Lie theory : open problems , 1998 .

[22]  Thomas Lam,et al.  Positroid varieties: juggling and geometry , 2009, Compositio Mathematica.

[23]  G. Lusztig.  Total positivity in partial flag manifolds , 1998 .

[24]  I. J. Schoenberg.  On Totally Positive Functions, LaPlace Integrals and Entire Functions of the LaGuerre-Polya-Schur Type. , 1947, Proceedings of the National Academy of Sciences of the United States of America.

[25]  L. Williams,et al.  The Totally Nonnegative Part of G/P is a CW Complex , 2008, 0802.0889.

[26]  M. Reineke.  Every Projective Variety is a Quiver Grassmannian , 2012, 1204.5730.

[27]  M. Finkelberg,et al.  Symplectic Degenerate Flag Varieties , 2011, Canadian Journal of Mathematics.

[28]  T. Lam.  Totally nonnegative Grassmannian and Grassmann polytopes , 2015, 1506.00603.

[29]  B. Keller,et al.  Desingularizations of quiver Grassmannians via graded quiver varieties , 2013, 1305.7502.

[30]  E. Feigin,et al.  Homological approach to the Hernandez-Leclerc construction and quiver varieties , 2013, 1302.5297.

[31]  David E. Speyer,et al.  Projections of Richardson Varieties , 2010, 1008.3939.

[32]  M. Finkelberg,et al.  Degenerate affine Grassmannians and loop quivers , 2014, 1410.0777.