A Study of the representations supported by the orbit closure of the determinant

TL;DR

This paper investigates the representations contained within the orbit closure of the determinant, linking their existence to the validity of the Latin Square Conjecture, thus connecting algebraic geometry with combinatorial conjectures.

Contribution

It establishes the existence of a large family of representations supported by the orbit closure of the determinant, contingent on the Latin Square Conjecture.

Findings

Existence of many representations supported by the orbit closure of the determinant

Connection between algebraic geometry and Latin Square Conjecture

Results depend on the validity of the Latin Square Conjecture

Abstract

We show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the celebrated `Latin Square Conjecture' due to Alon-Tarsi or more precisely on the validity of an equivalent `column Latin Square Conjecture' due to Huang-Rota.