Syzygies for the vector invariants of the dihedral group

TL;DR

This paper investigates the relations among invariants of the dihedral group acting on vector tuples, showing they are generated by relations involving at most three vectors, and provides minimal generating systems for specific cases.

Contribution

It introduces a new understanding of the generators of the $GL$-ideal of relations for dihedral group invariants, including minimal systems for key cases.

Findings

The $GL$-ideal is generated by relations with no more than 3 vectors.

Minimal generating systems are found for $m=2$ and for the dihedral group of order 8.

The results simplify the understanding of relations among dihedral group invariants.

Abstract

The problem of finding generators of the $GL$-ideal of the relations between the generators of the algebra of invariants of the dihedral group acting on $m$-tuples of vectors from its defining $2$-dimensional representation is studied. It is shown that this $GL$-ideal is generated by relations depending on no more than $3$ vector variables. A minimal $GL$-ideal generating system is found for the case when $m=2$, and for the case of the dihedral group of order $8$ and arbitrary $m$.