Lovász–Saks–Schrijver Ideal

• The LSS ideal is a quadratic ideal that encodes orthogonal vector assignments to graph vertices, reflecting the graph's structure via polynomial equations.
• It links algebraic properties such as primality, complete intersection, and radicality with combinatorial invariants like positive matching decomposition, providing explicit thresholds for various graph types.
• The construction extends naturally to hypergraphs and twisted cases, unifying the study of determinantal varieties, tensor sections, and orthogonal representations.

The Lovász–Saks–Schrijver (LSS) ideal is a quadratic ideal associated to graphs and hypergraphs, central in the algebraic study of orthogonal representations, determinantal varieties, and tensor sections. For a fixed graph $G$ and integer %%%%1%%%%, the LSS-ideal encodes the conditions for assigning $d$-vectors to vertices such that orthogonality constraints are fulfilled for specified pairs, directly reflecting graph-theoretic structure in polynomial equations. These ideals exhibit profound connections to combinatorial invariants, symmetric determinantal ideals, and paving matroids, and have been the focus of extensive research investigating their algebraic properties, primary decomposition, and geometric meaning.

1. Precise Definition

Let $G = ([n], E)$ be a finite simple graph and $d \geq 1$. In the polynomial ring

$$S = K[x_{i1}, x_{i2}, \ldots, x_{id} \mid i=1, \ldots, n],$$

define, for each unordered pair $\{i, j\}$,

$f_{ij} = \sum_{k=1}^d x_{ik} x_{jk} \in S.$

The $d$-dimensional Lovász–Saks–Schrijver ideal of $G$ is

$$L_G^{(d)} = (f_{ij} : \{i, j\} \notin E(G)) \subseteq S,$$

i.e., it is generated by quadratic forms for edges of the complement graph $\overline{G}$ (Conca et al., 2018). For hypergraphs, the construction generalizes with $f_e^{(d)} = \sum_{j=1}^d \prod_{i\in e} y_{ij}$ for each edge $e$ (Gharakhloo et al., 2022). The quotient ring $S/L_G^{(d)}$ parametrizes orthogonal representations of $\overline{G}$: assigning $v_i \in K^d$ to vertex $i$ such that $v_i \cdot v_j = 0$ for each $\{i, j\} \in E(\overline{G})$.

2. Geometric and Combinatorial Interpretation

A point in $\mathrm{Spec}\ S$ gives a $d$-vector assignment to each vertex, and the quadratic equations stipulate mutual orthogonality for prescribed non-edge pairs. This yields the affine algebraic variety: $$V(L_G^{(d)}) = \{ (v_1, \ldots, v_n) \in (K^d)^n : v_i \perp v_j \ \forall \{i, j\} \in E(\overline{G}) \}$$ which is the variety of all $d$-dimensional orthogonal representations of the complement graph. For forests and trees, this variety decomposes into irreducible components precisely indexed by $G$-admissible subsets, defined via matroidal ranks and combinatorial constraints (Liwski, 28 Dec 2025). For hypergraphs, LSS-ideals parametrize coordinate sections of symmetric tensor varieties (Gharakhloo et al., 2022).

3. Algebraic Properties and Invariants

Complete Intersection, Primality, and Radicality

For any graph $G$, the algebraic properties of $L_G^{(d)}$ are deeply governed by both graph-theoretic invariants and $d$:

• If $L_G^{(d)}$ is prime, it is a complete intersection; if it is a complete intersection, $L_G^{(d+1)}$ is prime (Conca et al., 2018, Gharakhloo et al., 2022).
• These properties hereditarily persist to all subgraphs and incrementally in $d$.
• Define the positive matching decomposition number

  $\mathrm{pmd}(G) = \min\{p \mid G \text{ admits a partition of the edge set into %%%%27%%%% positive matchings}\}.$

• For $d \geq \mathrm{pmd}(G)$, $L_G^{(d)}$ is radical and a complete intersection; for $d \geq \mathrm{pmd}(G) + 1$, it is prime. This yields explicit bounds, e.g., for bipartite graphs $\mathrm{pmd}(G) \leq n-1$ (Conca et al., 2018, Gharakhloo et al., 2022).

The Forest and Tree Case

Let $G$ be a forest on $n$ vertices with maximum degree $\Delta(G)$. Then for all $d \geq 1$ (Conca et al., 2018, Liwski, 28 Dec 2025, Nambi et al., 2023):

• $L_G^{(d)}$ is radical.
• $L_G^{(d)}$ is a complete intersection iff $d \geq \Delta(G)$.
• $L_G^{(d)}$ is prime iff $d \geq \Delta(G) + 1$.

For trees, the explicit Gröbner basis, Hilbert series, and Krull dimension via combinatorial formulas are established (Ghouchan et al., 2023).

4. Primary and Irreducible Decomposition

In the forest case, the irreducible components of $V(L_G^{(d)})$ are indexed by $G$-admissible subsets $S$, with $V_S$ defined by killing coordinates $x_{i,k}$ for $i \in S$, and joining with determinantal equations (Plücker-type minors) for dependent hyperplanes induced by the paving matroid $\mathcal{M}(G)$: $$L_G^{(d)} = \bigcap_{S \text{ admissible}} \left( (x_{i,k} : i \in S,\, 1 \leq k \leq d) + \sqrt{I_{G_{[n]\setminus S}}} \right)$$ where $I_{G_{[n]\setminus S}}$ is generated by determinantal substitutions for neighborhoods of size at least $d$ (Liwski, 28 Dec 2025). This decomposition is explicit, and all components are prime.

5. Connections to Determinantal, Tensor, and Frame Ideals

There is a canonical transfer of radicality, primeness, and CI property from $L_G^{(d)}$ to the ideal $I_{d+1}(X_G)$ of $(d+1)$-minors of the symmetric matrix with zero pattern determined by $G$ (Conca et al., 2018): $$K[X_G]/I_{d+1}(X_G) \cong K[Y Y^T]/(f_{ij} : \{i,j\} \in E(G)) = [K[x_{ik}]/L_G^{(d)}]^{SO_d}$$ Particular cases extend to Pfaffian (skew-symmetric) and bipartite versions, with all CI/primeness results preserved. For tensor varieties, the LSS-ideal for hypergraphs controls the irreducibility of coordinate sections of symmetric tensors of rank at most $d$ (Gharakhloo et al., 2022).

6. Homological and Additional Algebraic Properties

• Betti numbers and resolutions: For $d=2$, trees and odd unicyclic graphs admit explicit minimal free resolutions, with Betti numbers given by combinatorial formulas (Kumar, 2019).
• The symmetric algebra of LSS ideals is generated by commutativity and claw relations, corresponding respectively to pairs of edges and claws in the graph; the structure is typically linear except in almost CI cases (Kumar, 2019).
• For powers and almost CIs, projective dimension and regularity are calculated explicitly, with Cohen–Macaulayness of the associated graded ring and Rees algebra proved for almost CIs (Nambi et al., 2023).
• Koszul property: $S/L_G^{(d)}$ is Koszul if $|E(G)| \leq n d$ or $|E(G)| \geq n d + 1 - \binom{n}{2}$; in particular, if $G$ is a tree or unicyclic, $S/L_G^{(d)}$ is always Koszul (Nambi et al., 2023).

7. Generalizations: Hypergraph and Twisted Ideals

For hypergraphs, LSS-ideals generalize with generators $\sum_{j=1}^d \prod_{i\in e} y_{ij}$ for each edge $e$ (Gharakhloo et al., 2022). The CI and primality criteria transport via positive matching decompositions ($\mathrm{pmd}(H)$). Furthermore, twisted LSS-ideals $\tilde{L}_G(d)$, defined using pairs of variables to encode antisymmetric forms, admit CI property when the twisted positive matching decomposition invariant $\mathrm{tpmd}(G)$ is at most $d$ (Nambi et al., 2023).

Table: Key Combinatorial Invariants and Thresholds

Graph/Hypergraph Type	Invariant	CI iff	Prime iff
Forest	$\Delta(G)$	$d \geq \Delta(G)$	$d \geq \Delta(G) + 1$
General Graph	$\mathrm{pmd}(G)$	$d \geq \mathrm{pmd}(G)$	$d \geq \mathrm{pmd}(G) + 1$
Hypergraph	$\mathrm{pmd}(H)$	$d \geq \mathrm{pmd}(H)$	$d > \mathrm{pmd}(H)$
Twisted LSS-ideal	$\mathrm{tpmd}(G)$	$d \geq \mathrm{tpmd}(G)$	–

References and Further Reading

• "Lovasz-Saks-Schrijver ideals and coordinate sections of determinantal varieties" (Conca et al., 2018)
• "Lovász--Saks--Schrijver Ideals and the Irreducible Components of the Variety of Orthogonal Representations of a Graph" (Liwski, 28 Dec 2025)
• "Hypergraph LSS-ideals and coordinate sections of symmetric tensors" (Gharakhloo et al., 2022)
• "Gröbner basis and Krull dimension of Lovász-Saks-Sherijver ideal associated to a tree" (Ghouchan et al., 2023)
• "(Almost) Complete Intersection Lovász-Saks-Schrijver ideals and regularity of their powers" (Nambi et al., 2023)
• "Lovász–Saks–Schrijver ideals and parity binomial edge ideals of graphs" (Kumar, 2019)
• "The variety of orthogonal frames" (Casabella et al., 31 Dec 2025)

This research program interlaces combinatorics, commutative algebra, and algebraic geometry, yielding explicit links between graphical invariants and deep algebraic properties of associated ideals and varieties.