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Lovász--Saks--Schrijver Ideals and the Irreducible Components of the Variety of Orthogonal Representations of a Graph

Emiliano Liwski

TL;DR

This work resolves the irreducible and primary decompositions of the orthogonal representation variety $OR_d(\bar{G})$ and the Lovász–Saks–Schrijver ideals $L_G(d)$ for forest graphs in dimension $d\ge3$ over $\mathbb{C}$. The authors introduce a paving matroid $\mathcal{M}(G)$ and the auxiliary varieties $U(S)$ and $V_S$, establishing that $OR_d(\bar{G})$ decomposes into $V_S$ indexed by $G$-admissible subsets, with explicit dimension $\dim(V_S)=d(n-|S|)-|E(G_{[n]\setminus S})|$. They further derive the ideals of $V_S$, showing $\mathbb{I}(V_S)=(x_{i,j}: i\in S) + \sqrt{I_{G_{[n]\setminus S}}}$, and obtain the primary decomposition of $L_G(d)$ as $L_G(d)=\bigcap_S\bigl((x_{i,j}: i\in S, j\in [d]) + \sqrt{I_{G_{[n]\setminus S}}}\bigr)$. The results generalize known cases (e.g., $d=1,2$) and provide concrete descriptions for forests, including explicit analyses for star, caterpillar, and binary-tree families, linking algebraic, geometric, and matroid-theoretic perspectives in this combinatorial-algebraic setting.

Abstract

Given a finite simple graph $G$ and a positive integer $d$, one can associate to $G$ the Lovász--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 $\mathcal{M}(G)$.

Lovász--Saks--Schrijver Ideals and the Irreducible Components of the Variety of Orthogonal Representations of a Graph

TL;DR

This work resolves the irreducible and primary decompositions of the orthogonal representation variety and the Lovász–Saks–Schrijver ideals for forest graphs in dimension over . The authors introduce a paving matroid and the auxiliary varieties and , establishing that decomposes into indexed by -admissible subsets, with explicit dimension . They further derive the ideals of , showing , and obtain the primary decomposition of as . The results generalize known cases (e.g., ) and provide concrete descriptions for forests, including explicit analyses for star, caterpillar, and binary-tree families, linking algebraic, geometric, and matroid-theoretic perspectives in this combinatorial-algebraic setting.

Abstract

Given a finite simple graph and a positive integer , one can associate to the Lovász--Saks--Schrijver ideal , an ideal generated by quadratic polynomials coming from orthogonality conditions. The corresponding variety , denoted , is the variety of orthogonal representations of the complement graph : its points are maps from the vertex set of to that send adjacent vertices of to orthogonal vectors. In this paper we study the irreducible decomposition of and the primary decomposition of . Our main focus is the case in which is a forest. Under this assumption, we determine the irreducible components of , compute their dimensions, and describe their defining equations, thereby obtaining the primary decomposition of . The key ingredient is a matroid-theoretic framework in which we associate to every forest a paving matroid .
Paper Structure (18 sections, 18 theorems, 88 equations, 5 figures)

This paper contains 18 sections, 18 theorems, 88 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation.
  3. LSS ideals.
  4. Main question.
  5. Related work.
  6. Outline of the strategy.
  7. Preliminaries
  8. Irreducible decomposition of OR for forest graphs
  9. Variety $U_{G}$
  10. Decomposition into varieties $V_{S}$
  11. Matroid $\mathcal{M}(G)$
  12. Identifying the maximal varieties $V_{S}$
  13. Irreducible decomposition of $\textup{OR}_{d}(\overline{G})$
  14. Irreducible decomposition of OR for forest graphs
  15. k
  16. ...and 3 more sections

Key Result

Theorem (A)

Let $G$ be a forest on $[n]$. The irreducible decomposition of $\textup{OR}_{d}(\overline{G})$ is where the union ranges over all $G$-admissible subsets of $[n]$, that is, all subsets $S\subseteq [n]$ such that for every $i\in S$, as in Definition def: admissible.

Figures (5)

  • Figure 1: A forest graph.
  • Figure 2: The matroid $\mathcal{M}(G)$ from Example \ref{['example: description of MG']} associated with the forest in Figure \ref{['forest graph']}.
  • Figure 3: The star graph $K_{1,7}$.
  • Figure 4: A caterpillar tree.
  • Figure 5: A binary tree.

Theorems & Definitions (58)

  • Definition 1.1
  • Definition 1.2: Definition \ref{['def: US and VS']}
  • Definition 1.3: Definition \ref{['def: MG']}
  • Theorem (A)
  • Theorem (B)
  • Theorem (C)
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Example 2.4
  • ...and 48 more