Generic initial ideals of modular polynomial invariants

TL;DR

This paper investigates the structure of generic initial ideals of Hilbert ideals in modular invariant theory, providing explicit calculations for cyclic groups and insights into Borel fixed ideals for the Klein four group.

Contribution

It explicitly computes the generic initial ideals of Hilbert ideals for cyclic groups of prime order and analyzes their properties for the Klein four group, advancing understanding in modular invariant theory.

Findings

Explicit gin for cyclic groups of prime order under all monomial orders.

Hilbert ideals of the Klein four group are Borel fixed under certain variable orderings.

gin respects permutations of variables in the monomial order.

Abstract

We study the generic initial ideals (gin) of certain ideals that arise in modular invariant theory. For all cases an explicit generating set is known we calculate the generic initial ideal of the Hilbert ideal of a cyclic group of prime order for all monomial orders. We also clarify the Klein four group and note that its Hilbert ideals are Borel fixed with certain orderings of the variables. In all situations we consider, there is a monomial order such that the gin of the Hilbert ideal is equal to its initial ideal. Along the way we show that gin respects a permutation of the variables in the monomial order.