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| Main Author: | |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2111.06351 |
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Table of Contents:
- We construct moduli spaces of linear self-maps of projective space with marked points, up to projective equivalence. That is, we let the special linear group act simultaneously by conjugation on projective linear maps and diagonally on $(\mathbb{P}^N)^n$, and we take the geometric invariant theory (GIT) quotient. These moduli spaces arise in algebraic dynamics and integrable systems. Our main result is a dynamical characterization of the GIT semistable and stable loci in the space of linear maps $T$ with marked points. We show that GIT stability can be checked by counting the marked points on flags with certain Hessenberg functions relative to $T$. The proof is combinatorial: to describe the weight polytopes for this action, we compute the vertices and facets of certain convex polyhedra generated by roots of the $A_N$ lattice.