Permanent versus determinant: not via saturations

TL;DR

This paper investigates the algebraic structure of the orbit closure of the determinant polynomial, revealing that certain partitions are contained in its monoid, which impacts the understanding of obstructions in the permanent versus determinant problem.

Contribution

It proves that the saturation of the monoid of irreducible representations contains all partitions with length at most n and size divisible by n, clarifying the structure of representation-theoretic obstructions.

Findings

Saturation of S(Det_n) contains all partitions with length ≤ n and size divisible by n

Representation-theoretic obstructions are holes in the monoid S(Det_n)

Provides insight into the algebraic structure relevant to the permanent vs. determinant problem

Abstract

Let Det_n denote the closure of the GL_{n^2}(C)-orbit of the determinant polynomial det_n with respect to linear substitution. The highest weights (partitions) of irreducible GL_{n^2}(C)-representations occurring in the coordinate ring of Det_n form a finitely generated monoid S(Det_n). We prove that the saturation of S(Det_n) contains all partitions lambda with length at most n and size divisible by n. This implies that representation theoretic obstructions for the permanent versus determinant problem must be holes of the monoid S(Det_n).