Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph :
the Lovász-Saks-Schrijver ideal defining the -dimensional orthogonal representations of the graph complementary to , and
the determinantal ideal of the -minors of a generic symmetric matrix with in positions prescribed by the graph .
In characteristic these two ideals turn out to be closely related and algebraic properties such as being radical, prime or a complete intersection transfer from the Lovász–Saks–Schrijver ideal to the determinantal ideal. For Lovász–Saks–Schrijver ideals we link these properties to combinatorial properties of and show that they always hold for large enough. For specific classes of graphs, such a forests, we can give a complete picture and classify the radical, prime and complete intersection Lovász–Saks–Schrijver ideals.
"Lovász–Saks–Schrijver ideals and coordinate sections of determinantal varieties." Algebra Number Theory 13 (2) 455 - 484, 2019. https://doi.org/10.2140/ant.2019.13.455