Asymptotic Schur decomposition of Veronese syzygy functors

TL;DR

This paper investigates the asymptotic behavior of Schur functor decompositions of Veronese syzygy functors, revealing their increasing complexity as the degree grows, through plethysm analysis.

Contribution

It introduces an asymptotic framework for understanding the Schur decomposition of Veronese syzygies and addresses related questions by Ein and Lazarsfeld.

Findings

Schur functor decompositions become very rich as degree d increases

Asymptotic analysis of plethysms reveals growth in complexity

Provides new insights into the structure of Veronese syzygies

Abstract

The syzygies of the d-th Veronese embedding of $\mathbb P(V)$ are functors of the complex vector space V. From a certain perspective, we show that as d grows, their Schur functor decomposition is very rich whenever they are not zero. This is deduced from an asymptotic study of related plethysms. We also obtain other results related to a question of Ein and Lazarsfeld.