Standard monomials and invariant theory for arc spaces I: general linear group

TL;DR

This paper develops a standard monomial basis for arc spaces of determinantal varieties and establishes invariant theory results for the general linear group in this context.

Contribution

It introduces a standard monomial basis for arc spaces of determinantal varieties and proves the invariant theory theorems for the general linear group on these spaces.

Findings

Constructed a standard monomial basis for arc spaces of determinantal varieties.

Proved the arc space analogue of the fundamental theorems of invariant theory for GL.

Extended invariant theory to the setting of arc spaces.

Abstract

This is the first in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field $K$, we construct a standard monomial basis for the arc space of the determinantal variety over $K$. As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the general linear group.