Degrees of regular sequences with a symmetric group action

TL;DR

This paper investigates the degrees of elements in symmetric group-stable regular sequences in polynomial rings, providing criteria for their existence based on prior classification of their isomorphism types.

Contribution

It extends previous classification work by analyzing the degrees of elements in such regular sequences and establishing existence criteria for specific degrees.

Findings

Criteria for the existence of regular sequences in certain degrees.

Classification of possible degrees based on isomorphism types.

Insights into the structure of symmetric group-stable ideals.

Abstract

We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees.