Lovász--Saks--Schrijver Ideals and the Irreducible Components of the Variety of Orthogonal Representations of a Graph
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| Format: | Preprint |
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2025
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| _version_ | 1866914223114682368 |
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| author | Liwski, Emiliano |
| author_facet | Liwski, Emiliano |
| contents | 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 $\mathcal{M}(G)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22954 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lovász--Saks--Schrijver Ideals and the Irreducible Components of the Variety of Orthogonal Representations of a Graph Liwski, Emiliano Combinatorics Commutative Algebra Algebraic Geometry 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 $\mathcal{M}(G)$. |
| title | Lovász--Saks--Schrijver Ideals and the Irreducible Components of the Variety of Orthogonal Representations of a Graph |
| topic | Combinatorics Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2512.22954 |