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Subspaces fixed by a nilpotent matrix

Marvin Anas Hahn, Gabriele Nebe, Mima Stanojkovski and Bernd Sturmfels

Vol. 1 (2024), No. 1, 1–15
Abstract

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

Keywords
nilpotent matrices, subspaces, shuffle equations, affine Grassmannian
Mathematical Subject Classification
Primary: 14M15
Secondary: 13P10, 68W30
Milestones
Received: 2 July 2022
Revised: 9 March 2023
Accepted: 9 March 2023
Published: 26 November 2023
Authors
Marvin Anas Hahn
School of Mathematics
Trinity College Dublin
Ireland
Gabriele Nebe
Lehrstuhl fuer Algebra und Zahlentheorie
RWTH Aachen University
Germany
Mima Stanojkovski
Dipartimento di Matematica
Università di Trento
Italy
Bernd Sturmfels
Max Planck Institute for Mathematics in the Sciences
Leipzig
Germany
University of California at Berkeley
California
United States