Lovasz-Saks-Schrijver ideals and coordinate sections of determinantal varieties

TL;DR

This paper explores algebraic properties of two graph-associated ideals, revealing their close relationship and linking algebraic features to graph combinatorics, especially for large dimensions and specific graph classes.

Contribution

It establishes a connection between Lovasz-Saks-Schrijver and determinantal ideals, providing classifications and conditions for algebraic properties based on graph structure.

Findings

Algebraic properties transfer between the two ideals in characteristic 0.

Properties like being radical, prime, or a complete intersection depend on graph and dimension.

Complete classifications for forests and large dimensions.

Abstract

Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph G: --> the Lovasz-Saks-Schrijver ideal defining the d-dimensional orthogonal representations of the graph complementary to G and --> the determinantal ideal of the (d+1)-minors of a generic symmetric with 0s in positions prescribed by the graph G. In characteristic 0 these two ideals turns out to be closely related and algebraic properties such as being radical, prime or a complete intersection transfer from the Lovasz-Saks-Schrijver ideal to the determinantal ideal. For Lovasz-Saks-Schrijver ideals we link these properties to combinatorial properties of G and show that they always hold for d large enough. For specific classes of graph, such a forests, we can give a complete picture and classify the radical, prime and complete intersection Lovasz-Saks-Schrijver ideals.