Lov\'asz--Saks--Schrijver Ideals and the Irreducible Components of the Variety of Orthogonal Representations of a Graph

Abstract

Given a finite simple graph $G$ and a positive integer $d$, one can associate to $G$ the Lov\'asz--Saks--Schrijver ideal $L_{G}(d)$, an ideal generated by quadratic polynomials coming from orthogonality conditions. The corresponding variety $\mathbb{V}(L_{G}(d))$, denoted $\mathrm{OR}_{d}(\overline{G})$, is the variety of orthogonal representations of the complement graph $\overline{G}$: its points are maps from the vertex set of $G$ to $\mathbb{K}^{d}$ that send adjacent vertices of $G$ to orthogonal vectors. In this paper we study the irreducible decomposition of $\mathrm{OR}_{d}(\overline{G})$ and the primary decomposition of $L_{G}(d)$. Our main focus is the case in which $G$ is a forest. Under this assumption, we determine the irreducible components of $\mathrm{OR}_{d}(\overline{G})$, compute their dimensions, and describe their defining equations, thereby obtaining the primary decomposition of $L_{G}(d)$. The key ingredient is a matroid-theoretic framework in which we associate to every forest $G$ a paving matroid $\mathcal{M}(G)$.