Subspaces Fixed by a Nilpotent Matrix

arXiv (Cornell University)

Abstract

The linear spaces that are fixed by a given nilpotent $n \times n$ matrix form a subvariety of the Grassmannian. We classify these varieties for small $n$. Mutiah, Weekes and Yacobi conjectured that their radical ideals are generated by certain linear forms known as shuffle equations. We prove this conjecture for $n \leq 7$, and we disprove it for $n=8$. The question remains open for nilpotent matrices arising from the affine Grassmannian.

Authors 4

1. Marvin Anas Hahn Aachen

   University of Trento · Trinity College Dublin · RWTH Aachen University · University of California, Berkeley

   Affiliation as printed

   Trinity College Dublin RWTH Aachen Mima Stanojkovski , Università di Trento MPI-MiS Leipzig and UC Berkeley

2. Gabriele Nebe Aachen

   University of Trento · Trinity College Dublin · RWTH Aachen University · University of California, Berkeley

   Affiliation as printed

   Trinity College Dublin RWTH Aachen Mima Stanojkovski , Università di Trento MPI-MiS Leipzig and UC Berkeley

3. Mima Stanojkovski Aachen

   University of Trento · Trinity College Dublin · RWTH Aachen University · University of California, Berkeley

   Affiliation as printed

   Trinity College Dublin RWTH Aachen Mima Stanojkovski , Università di Trento MPI-MiS Leipzig and UC Berkeley

4. Bernd Sturmfels Aachen

   University of Trento · Trinity College Dublin · RWTH Aachen University · University of California, Berkeley

   Affiliation as printed

   Trinity College Dublin RWTH Aachen Mima Stanojkovski , Università di Trento MPI-MiS Leipzig and UC Berkeley