Geometry of orbits of permanents and determinants

TL;DR

This paper investigates the geometric properties of the orbit closures of the determinant and permanent polynomials, revealing their non-normality and contributing to the understanding of their algebraic structure.

Contribution

It establishes that the orbit closure of the determinant is not normal and extends this result to the permanent multiplied by a linear form.

Findings

Orbit closure of the determinant is not normal.

Orbit closure of the permanent times a linear form is not normal.

Advances understanding of algebraic geometry of polynomial orbits.

Abstract

We prove that the orbit closure of the determinant is not normal. A similar result is obtained for the orbit closure of the permanent multiplied by a power of a linear form.