On the ideal of orthogonal representations of a graph in R2R2

In this paper, we study orthogonal representations of simple graphs G   in RdRd from an algebraic perspective in case d=2d=2. Orthogonal representations of graphs, introduced by Lovász, are maps from the vertex set to RdRd where non-adjacent vertices are sent to orthogonal vectors. We exhibit algebraic properties of the ideal generated by the equations expressing this condition and deduce geometric properties of the variety of orthogonal embeddings for d=2d=2 and RR replaced by an arbitrary field. In particular, we classify when the ideal is radical and provide a reduced primary decomposition if −1∉K. This leads to a description of the variety of orthogonal embeddings as a union of varieties defined by prime ideals. In particular, this applies to the motivating case K=RK=R.