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

TL;DR

This paper analyzes the structure of the variety of orthogonal representations of a graph's complement, providing a detailed decomposition and equations for forests using matroid theory.

Contribution

It determines the irreducible components and primary decomposition of Lovász--Saks--Schrijver ideals for forests, linking graph theory and matroid theory.

Findings

Irreducible components of the variety are explicitly characterized.

Dimensions and defining equations of components are computed.

Primary decomposition of the ideal is obtained for forest graphs.

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 $\mathcal{M}(G)$.