Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph G: • the Lovász-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 matrix with 0 in positions prescribed by the graph G. In characteristic 0 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 G and show that they always hold for d 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
Conca A.;
2019-01-01
Abstract
Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph G: • the Lovász-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 matrix with 0 in positions prescribed by the graph G. In characteristic 0 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 G and show that they always hold for d 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.File | Dimensione | Formato | |
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(2019) Conca Welker Lovász–Saks–Schrijver ideals and coordinate sections of determinantal ideals.pdf
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