Syzygies for the vector invariants of the dihedral group

Abstract

The dihedral group D2n of order 2n acts on the space of m-tuples of vectors from its defining 2-dimensional representation. The corresponding algebra of polynomial invariants has a natural structure as a module over the general linear group GLm(C). Therefore the ideal of relations between the generators of the algebra of invariants can be treated as a GL-ideal (i.e. an ideal stable with respect to the appropriate action of GLm(C)). 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 cases of m=2 and an arbitrary n, and for the cases of an arbitrary m and n=4, n=5, and n=6.