Relations between the minors of a generic matrix

TL;DR

This paper investigates the algebraic relations among minors of a fixed size in a generic matrix, identifying minimal relations and proposing a conjecture on the generating system of the ideal of relations using representation theory.

Contribution

It introduces new minimal relations among minors of fixed size and provides evidence supporting a conjecture about the generating system of their ideal, expanding understanding beyond maximal minors.

Findings

Minimal relations in degrees 2 and 3 identified.

Evidence supports the conjecture on the generating system of relations.

Representation theory approach used to analyze relations.

Abstract

It is well-known that the Pl\"ucker relations generate the ideal of relations of the maximal minors of a generic matrix. In this paper we discuss the relations between minors of a (non-maximal) fixed size. We will exhibit minimal relations in degrees 2 (non-Pl\"ucker in general) and 3, and give some evidence for our conjecture that we have found the generating system of the ideal of relations. The approach is through the representation theory of the general linear group.